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Volume 243 No. 1 November 2009

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PACIFIC JOURNAL OF MATHEMATICSVol. 243, No. 1, 2009

DOUBLE AFFINE LIE ALGEBRAS AND FINITE GROUPS

NICOLAS GUAY, DAVID HERNANDEZ AND SERGEY LOKTEV

We begin to study the Lie theoretical analogues of symplectic reflectionalgebras for a finite cyclic group 0; we call these algebras “cyclic doubleaffine Lie algebras”. We focus on type A: In the finite (respectively affine,double affine) case, we prove that these structures are finite (respectivelyaffine, toroidal) type Lie algebras, but the gradings differ. The case thatis essentially new is sln(C[u, v] o 0). We describe its universal central ex-tensions and start the study of its representation theory, in particular of itshighest weight integrable modules and Weyl modules. We also consider thefirst Weyl algebra A1 instead of the polynomial ring C[u, v], and, more gen-erally, a rank one rational Cherednik algebra. We study quasifinite highestweight representations of these Lie algebras.

1. Introduction 12. Matrix Lie algebras over rings 43. Cyclic affine Lie algebras 54. Cyclic double affine Lie algebras 105. Representations of cyclic double affine Lie algebras 156. Matrix Lie algebras over rational Cherednik algebras of rank

one 247. Highest weight representations for matrix Lie algebras over

Cherednik algebras of rank one 308. Further discussions 36Acknowledgments 38References 38

1. Introduction

Double affine Hecke algebras have been well studied for more than fifteen yearsnow, although they are still very mysterious, and symplectic reflection algebrasappeared over seven years ago [Etingof and Ginzburg 2002] as generalizations of

MSC2000: 17B67.Keywords: affine Lie algebras, toroidal Lie algebras, symplectic reflection algebras.Hernandez is supported partially by ANR through project “Géométrie et Structures AlgébriquesQuantiques”.

1

http://pjm.berkeley.edu

http://dx.doi.org/10.2140/pjm.2009.243-1

2 NICOLAS GUAY, DAVID HERNANDEZ AND SERGEY LOKTEV

double affine Hecke algebras of rational type. Even more mysterious are the doubleaffine Lie algebras and their quantized version introduced in [Ginzburg et al. 1995]and studied for instance in [Hernandez 2005; 2007; Nagao 2007; Nakajima 2002;Schiffmann 2006; Varagnolo and Vasserot 1996; 1998] and the references in thesurvey [Hernandez 2009].

In this paper, we study candidates for Lie theoretical analogues of symplecticreflection algebras, which we call “cyclic double affine Lie algebras”. We look at afamily of Lie algebras which have a lot of similarities with affine and double affineLie algebras, but whose structure depends on a finite cyclic group 0.

More precisely, we will be interested in the Lie algebras

sln(C[u]o0), sln(C[u, v]o0), sln(C[u±1, v]o0),

sln(C[u±1]o0), sln(C[u, v]0), sln(C[u±1, v±1

]o0),

sln(A1 o 0) (where A1 is the first Weyl algebra), sln(Ht,c(0)) (where Ht,c(0)

is a rank-one rational Cherednik algebra) and their universal central extensions.This is motivated by the recent work [Guay 2009b], in which deformations of theenveloping algebras of some Lie algebras closely related to these were constructedand connected to symplectic reflection algebras for wreath products via a functorof Schur–Weyl type. When 0 is trivial, such deformations in the case of C[u±1

]

are the affine quantum groups, whereas the case C[u] corresponds to Yangians. Inthe double affine setup, the quantum algebras attached to C[u±1, v±1

],C[u±1, v]

and C[u, v] for sln are the quantum toroidal algebras, the affine Yangians and thedeformed double current algebras [Guay 2005; 2007].

In this article, we want to study more the structure and representation theory forthe Lie algebras above, hoping that, in a future work, we will be able to extendsome of our results to the deformed setup. We consider the central extensions fora number of reasons. In the affine case, the full extent of the representation theorycomes to life when the center acts not necessarily trivially. Certain presentationsof those Lie algebras are actually simpler to state for central extensions since theyinvolve fewer relations; for example, some of the results can be extended withoutmuch difficulty to those central extensions. As vector spaces, the centers of theuniversal extensions are given by certain first cyclic homology groups.

At first sight, one may be tempted to think that introducing the group 0 leadsto Lie algebras that are different from those that have interested Lie theorists sincethe advent of Kac–Moody Lie algebras (it was our first motivation), but this isnot entirely the case. Indeed, in the one variable case, when we consider not onlyordinary polynomials but Laurent polynomials, we prove that we get back affineLie algebras (Proposition 3.8); this agrees with conjectures of V. Kac [1968] andthe classification obtained by V. Kac [1968; 1990] and O. Mathieu [1986; 1992].

DOUBLE AFFINE LIE ALGEBRAS AND FINITE GROUPS 3

In the case of Laurent polynomials in two variables, we recover toroidal Lie al-gebras (Proposition 4.1). (The mixed case C[u±1, v] also does not yield new Liealgebras.) However, when we consider only polynomials in nonnegative powers ofthe variables, we obtain distinctly new Lie algebras (see Proposition 5.3).

Another motivation comes from geometry. The loop algebra sln(C[u±1]) can

be viewed as the space of polynomial maps C×→ sln . One can also consider theaffine line instead of the torus C×, or, more generally, the space of regular mapsX→ sln , where X is an arbitrary affine algebraic variety [Feigin and Loktev 2004].When X is two-dimensional, the most natural candidate is the torus C× × C×,although a simpler case is the plane C2. The variety X does not necessarily haveto be smooth and one interesting singular two-dimensional case is provided by theKleinian singularities C2/G, where G is a finite subgroup of SL2(C). We are thusled to the problem of studying the Lie algebras sln(C[u, v]G), where C[u, v]G isthe ring of invariant elements for the action of G. However, following one of themain ideas explained in the introduction of [Etingof and Ginzburg 2002], it maybe interesting to replace C[u, v]G by the smash product C[u, v]o G. Moreover,we can expect the full representation theory to come to life when we consider theuniversal central extensions of sln(C[u, v]G) and of sln(C[u, v]o G). Feigin andLoktev [2004] showed in the case of a smooth affine variety X that the dimensionof the local Weyl modules at a point p does not depend on p. One goal is tounderstand Weyl modules supported at a Kleinian singularity.

This paper is organized as follows. We will denote by 0 the group Z/dZ,whereas G will be a more general finite group. After general reminders on matrixLie algebra over rings (in particular with the example of sln(C[G]) in Section 2, westart with the affine case in Section 3. We study the structure of sln(C[u]o0) andsln(C[u±1

]o 0), obtain different types of decomposition, and give presentationsin terms of generators and relations. We prove that sln(C[u±1

]o0) is simply theusual loop algebra slnd(C[t±1

]), but with a nonstandard grading. Guided by theaffine setup, we prove analogous results for the double affine cases in Section 4,the representations being studied in Section 5. We consider certain highest weightmodules for

sln(C[u, v]o0), sln(C[u±1, v]o0), sln(C[u±1, v±1]o0)

and state a criterion for the integrability of their unique irreducible quotients. Wealso study some Weyl modules for sln(C[u, v] o G) and sln(C[u, v]0). In thefirst case, we show that, contrary to what might be expected at first sight, Weylmodules are rather trivial; in the second case, we can apply results of Feigin andLoktev to derive formulas for the dimension some of the local Weyl modules, andwe establish a lower bound on their dimensions. In Section 6, assuming usually thatt 6= 0, we study parabolic subalgebras of gln(Ht,c(0)) and construct an embedding


of this Lie algebra into a Lie algebra of infinite matrices. This is useful in Section 7for constructing quasifinite highest weight modules. The main result of this sectionis a criterion for the quasifiniteness of the irreducible quotients of Verma modules.Further possible directions of research are discussed in Section 8.

2. Matrix Lie algebras over rings

2A. General results. In this section, we present general definitions and results thatwill be useful later. All algebras and tensor products are over C unless specifiedotherwise.

Definition 2.1. Let A be an arbitrary associative algebra. The Lie algebra sln(A)is defined as the derived Lie algebra [gln(A), gln(A)], where gln(A)= gln ⊗ A.

In other words, the Lie subalgebra sln(A) ⊂ gln(A) is the sum of sln(C)⊗ Aand of the space of all scalar matrices with entries in [A, A]. Thus the cyclichomology group HC0(A)= A/[A, A] accounts for the discrepancy between sln(A)and gln(A).

Since sln(A) is a perfect Lie algebra (that is, [sln(A), sln(A)] = sln(A)), it hasa universal central extension sln(A) unique up to isomorphism. The followingtheorem gives a simple presentation of sln(A) in terms of generators and relations.

Theorem 2.2 [Kassel and Loday 1982]. If n ≥ 3, then sln(A) is isomorphic to theLie algebra generated by elements Fi j (a), for 1 ≤ i, j ≤ n and a ∈ A, that satisfythe relations

[Fi j (a1), F jk(a2)] = Fik(a1a2) for i 6= j 6= k 6= i,

[Fi j (a1)Fkl(a2)] = 0, for i 6= j 6= k 6= l 6= i.

Here i 6= j 6= k 6= i means that i 6= j and j 6= k and k 6= i . We will use thisconvention in this paper.

When n = 2, one has to add generators H12(a1, a2) given by H12(a1, a2) =

[F12(a1), F21(a2)] for a1, a2 ∈ A, and the relations

[H12(a1, a2), F12(a3)] = F12(a1a2a3+ a3a2a1),

[H12(a1, a2), F21(a3)] = −F21(a3a1a2+ a2a1a3).

Kassel and Loday [1982] also prove that the center of sln(A) is isomorphic, as avector space, to the first cyclic homology group HC1(A). For G a finite group andA = C[G], we have HC1(A) = 0, but in the double affine case below, the centerwill be infinite-dimensional.

The following formulas taken from [Varagnolo and Vasserot 1998] help oneto understand better the bracket on sln(A). It is generally difficult to computeexplicitly the bracket with respect to the decomposition sln(A)∼= sln(A)⊕HC1(A),


but on can get some nice formulas by using a different splitting of sln(A). Let〈A, A〉 be the quotient of A⊗A by the two-sided ideal generated by a1⊗a2−a2⊗a1

and a1a2⊗ a3− a1⊗ a2a3− a2⊗ a3a1. The first cyclic homology group HC1(A)is, by definition, the kernel of the map 〈A, A〉� [A, A], a1⊗ a2 7→ [a1, a2].

For m1,m2 ∈ sln , a1, a2 ∈ A, and ( · , · ) the Killing form on sln , set

[m1,m2]+ = m1m2+m2m1−2n (m1,m2)I and [a1, a2]+ = a1a2+ a2a1.

Proposition 2.3 [Varagnolo and Vasserot 1998]. The Lie algebra sln(A) is isomor-phic to the vector space sln ⊗ A⊕〈A, A〉 endowed with the bracket

[m1⊗ a1,m2⊗ a2] =1n (m1,m2)〈a1, a2〉+

12 [m1,m2]⊗ [a1, a2]+

+12 [m1,m2]+⊗[a1, a2],

[〈a1, a2〉, 〈b1, b2〉] = 〈[a1, a2], [b1, b2]〉,

[〈a1, a2〉,m1⊗ a3] = m1⊗[[a1, a2], a3].

2B. Example: Special linear Lie algebras over group rings. Let G be a finitegroup. One interesting case for us is the group algebra A = C[G], in which caseHC0(A)∼= C⊕ cl(G), where cl(G) is the number of conjugacy classes of G.

Lemma 2.4. The Lie algebra sln(C[G]) is semisimple of Dynkin type A.

Proof. Recall that cl(G) is also the number of irreducible representations of G.Enumerate the irreducible representations of G by ρ1, . . . , ρcl(G), and let d( j)be the dimension of ρ j . Wedderburn’s theorem says that, as algebras, C[G] isisomorphic to

⊕cl(G)j=1 Md( j), where Md( j) is the associative algebra of d( j)×d( j)

matrices. We conclude that sln(C[G])∼=⊕cl(G)

j=1 slnd( j). �

Remark 2.5. The direct sum above is a direct sum of Lie algebras, that is, twodifferent copies of sln commute. A nondegenerate symmetric invariant bilinearform κ on the semisimple Lie algebra sln(C[G]) is given by the formula

κ(m1γ1,m2γ2)= Tr(m1 ·m2)δγ1=γ−12

for m1,m2 ∈ sln and γ1, γ2 ∈ G.

3. Cyclic affine Lie algebras

For an arbitrary ring R with action of a finite group G, the ring RoG is the span ofelements ag for a ∈ R and g ∈G with the relations (a1g1) ·(a2g2)= a1g1(a2)g1g2.In the previous section, just by considering group rings (over C), we ended upwith semisimple Lie algebras. Here, when R is a Laurent polynomial ring, onecan expect to obtain affine Kac–Moody algebras, which is indeed what happens.

3A. Definition and decomposition. Let ξ be a generator of 0 and ζ a primitive d-th root of unity. Let A=C[u±1

]o0 and B =C[u]o0. The action of 0 is definedby ξ(u) = ζu. We will be interested in the structure of the Lie algebra sln(A)


and of its universal central extension sln(A). We will also say a few words aboutsln(B). We will show that sln(A) is a graded simple Lie algebra and explain how itis related to the classification of such Lie algebras obtained by V. Kac [1968; 1990]and O. Mathieu [1986; 1992].

In the following, ≡ is the equivalence modulo d .

Lemma 3.1. [A, A] =d−1⊕i=1

C[u±1]ξ i⊕

⊕j∈Z, j 6≡0

C · u j ,

[B, B] =d−1⊕i=1

uC[u]ξ i⊕

⊕jZ≥1, j 6≡0

C · u j .

Proof. If 1≤ i ≤ d−1, then u jξ i= [u, u j−1ξ i

]/(1− ζ i ). If j ∈ Z and j 6≡ 0, thenu j= [ξ, u jξ−1

]/(ζ j− 1). This proves ⊇. Consider

[ukξa, umξ b] = (ζ am

− ζ bk)uk+mξa+b

and suppose that the right-hand side is in C[u±1]0. Then k+ l ≡ 0 and a+ b ≡ 0,

so ζ am− ζ bk

= 0. This proves ⊆. �

Corollary 3.2. HC0(B)∼= C[u]0 ⊕C[0].

The Lie algebra sln(A) admits different vector space decompositions, similarto the two standard triangular decompositions of affine Lie algebras. Let n+ (re-spectively n−) be the Lie algebra of strictly upper (respectively lower) triangularmatrices in sln (over C), and let h be the usual Cartan subalgebra of sln . Theelementary matrices in gln will be denoted Ei j and I will stand for the identitymatrix. In the following, by abuse of notation, an element g⊗ a will be denotedga for g ∈ gln and a ∈ A. We have the vector space isomorphisms (triangulardecompositions)

sln(A)∼= (n−A)⊕ (hA⊕ I [A, A])⊕ (n+A),

and sln(A) is also isomorphic to the sum(sln u−1C[u−1

]o0⊕

( ⊕j≤−1, j 6≡00≤i≤d−1

CI u jξ i⊕

⊕1≤i≤d−1

I u−dC[u−d]ξ i

)⊕n−C[0]

)

⊕

(hC[0]⊕

⊕1≤i≤d−1

CI ξ i)

⊕

(slnuC[u]o0⊕

( ⊕j≥1, j 6≡0

0≤i≤d−1

CI u jξ i⊕

⊕1≤i≤d−1

I udC[ud]ξ i

)⊕n+C[0]

).


These lead to similar decompositions for sln(B). These triangular decompositionsare similar to those considered, for instance, in [Khare 2009].

The first triangular decomposition is analogous to the loop triangular decom-position of affine Lie algebras, but in our situation the middle Lie algebra is notcommutative. The second triangular decomposition is similar to the decompositionof affine Lie algebras adapted to Chevalley–Kac generators, and it is of particularimportance as the middle term H in commutative. So the role of the Cartan sub-algebra will be played by the commutative Lie algebra H and our immediate aimis to obtain a corresponding appropriate root space decomposition of sln(A).

It will be convenient to work with the primitive idempotents of 0, so let us sete j =

1d

∑d−1i=0 ζ

−i jξ i . A vector space basis of H is given by

Hi, j =

{(Ei,i − Ei+1,i+1)e j for 1≤ i ≤ n− 1 and 0≤ j ≤ d − 1,En,ne j − E1,1e j+1 for i = 0 and 0≤ j ≤ d − 2.

We note that we could define H0,d−1 in the same way, but then we would get∑n−1i=0

∑d−1j=0 Hi, j = 0 in sln(A) (but lifts to a nonzero central element in sln(A)).

Lemma 3.3. A basis of the eigenspaces for nonzero eigenvalues for the adjointaction of H on sln(A) (except for H itself ) is given by the vectors

Ei j ukel for 1≤ i 6= j ≤ n, k ∈ Z, 0≤ l ≤ d − 1,

Ei i ukel for 1≤ i ≤ n, k 6≡ 0, 0≤ l ≤ d − 1.

Proof. This is a consequence of some simple computations: For

1≤ i 6= j ≤ n, 0≤ a ≤ n− 1, k ∈ Z, 0≤ l ≤ d − 1, 0≤ b ≤ d − 1,

we have

[Ha,b, Ei j ukel]=

(δb≡l(δa+1≡ j − δa≡ j )+ δb−k≡l(δa≡i − δa+1≡i )

)Ei j ukel

if a 6= 0,

(δn≡iδb−k≡l − δn≡ jδb≡l − δ1≡iδb+1−k≡l − δ j≡1δb+1≡l)Ei j ukel

if a = 0.

For 1≤ i ≤ n, k 6≡ 0 and 0≤ l ≤ d − 1, we have

[Ha,b, Ei i ukel] =

(δa≡i − δa+1≡i )(δb−k≡l − δb≡l)Ei i ukel if a 6= 0,(δn≡i (δb−k≡l − δb≡l)− δi≡1(δb+1−k≡l − δb+1≡l)

)Ei i ukel

if a = 0,

which proves the lemma. �


3B. Derivation element and imaginary roots. We will introduce the real roots,imaginary roots, and root spaces after adding a derivation d to sln(A).

The center of the universal central extension sln(A) of sln(A) is isomorphic toHC1(A); see [Kassel and Loday 1982]. It is known that HC1(A) is isomorphic to(C[u±1

]du)0/d(C[u±1]0), that is, the quotient of the space of 0-invariant 1-forms

on C× by the space of exact 1-forms coming from0-invariant Laurent polynomials.This can be deduced from the isomorphism A∼=Md(C[t±1

])— see Proposition 3.8.Thus this cyclic homology group is one-dimensional, with basis given by u−1du,which we denote as usual by c.

Definition 3.4. The cyclic affine Lie algebra sln(A) is obtained from sln(A) byadding a derivation d that satisfies the relations [d, Ei j ukel] = k Ei j ukel .

Set H=H⊕C·c⊕C·d. We can now introduce the roots as appropriate elementsof H∗0 = {λ ∈ H∗ | λ(c)= 0}. The real root spaces are spanned by the root vectorsEi j ukel , where 0≤ l ≤ d−1, 1≤ i, j ≤ n and k ∈Z, with the condition that k 6≡ 0if i = j .

The imaginary root spaces are spanned by the root vectors

Hi ukdel for 1≤ i ≤ n− 1, k 6= 0, 0≤ l ≤ d − 1,

Ennukdel − E11ukdel+1 for k 6= 0, 0≤ l ≤ d − 2.

We want to identify the root lattice as a lattice in H∗0. Let us introduce theelements εi,l ∈ H∗0 for 1≤ i ≤ n and 0≤ l ≤ d − 1 by setting

εi,l(Ha,b)= (δa=i − δa+1=i )δb≡l, εi,l(H0,b)= δn=iδb≡l − δi=1δb+1≡l,

and εi,l(d)= 0.

Definition 3.5. The real roots are εi,k+l − ε j,l + kδ for 1≤ i, j ≤ n, 0≤ l ≤ d−1and k ∈ Z with i 6= j or, if i = j , then k ≡ 0; the imaginary ones are kdδ, whereδ(d)= 1, δ(Ha,b)= 0 and k 6= 0.

They generate a lattice — the root lattice — in H∗0. As one can verify, the real rootspaces all have dimension one.

Lemma 3.6. The root lattice is freely generated by the roots

εi,l − εi+1,l for 1≤ i ≤ n− 1 and 0≤ l ≤ d − 1,

εn,l − ε1,l+1 for 0≤ l ≤ d − 2,

(εn,d−1− ε1,0)+ δ;

we call these the positive simple roots.


Proof. Note that

δ = ((εn,d−1− ε1,0)+ δ)+

d−1∑l=0

n−1∑i=1

(εi,l − εi+1,l)+

d−2∑l=0

(εn,l − ε1,l+1)

The set of simple roots contains nd elements, which is also dim H∗0. �

3C. Cyclic affine Lie algebras and affine Lie algebras.

Proposition 3.7. The Lie algebra sln(A) is graded simple (that is, it contains nonontrivial graded ideal).

Proof. Suppose that I =∑

m∈Z Im is a nonzero graded ideal of sln(C[u±1]o 0).

Then I is stable under the adjoint action of H; hence each graded piece Im mustdecompose into the direct sum of all the root spaces contained in Im . It can bechecked that the ideal generated by any real root vector is the whole sln(A), so if Icontains a real root vector, then I is the whole Lie algebra. Moreover, if I containsan imaginary root vector, then it contains also a real one. �

However, the Lie algebra sln(A) is not simple if we do not take the gradinginto account, as can be seen from Proposition 3.8 below. A conjecture of V. Kac[1968], proved in general by O. Mathieu [1986; 1992], gives a classification ofgraded simple Lie algebras of polynomial growth according to which, followingProposition 3.7, sln(A) must be isomorphic to a (perhaps twisted) loop algebra.This is indeed the case, although the isomorphism does not respect the naturalgrading on the loop algebra slnd ⊗C[t±1

].

Proposition 3.8. The Lie algebras sln(A) and slnd(C[t±1]) are isomorphic.

Proof. An isomorphism is given explicitly by the following formulas: If

0≤ l ≤ d − 1, k ∈ Z, −l ≤ r ≤ d − l − 1, 1≤ i 6= j ≤ n,

then

En(l+r)+i,nl+ j tk↔ Ei j ukd+r el,

En(l+r)+i,nl+i tk↔ Ei i ukd+r el for r 6= 0,

(Enl+i,nl+i − Enl+i+1,nl+i+1)tk↔ Hi ukdel for i 6= n,

(Enl,nl − Enl+1,nl+1)tk↔ Ennukdel − E11ukdel+1 for l 6= 0.

These formulas can be obtained via the algebra isomorphism A∼=Md(C[t±1]) that

is given by El+r,l tk↔ ukd+r el . �

Let φ : slnd(C[t±1]) ∼−→ sln(C[u±1

]o 0) be the isomorphism in the proof ofProposition 3.8. We can put a grading on slnd by giving Ei j degree j − i . Thisinduces another grading on slnd(C[t±1

]) besides the one coming from the powersof t ; the total of these two is the grading given by deg(Ei j (tr ))= j − i + r .


Proposition 3.9. φ is an isomorphism of graded Lie algebras when slnd(C[t±1])

is endowed with the total grading and sln(C[u±1]o0) is graded by powers of u.

Let p be the parabolic subalgebra of slnd consisting of all the lower triangularmatrices and the matrices with d blocks of n×n matrices along the diagonal. Let n

be the nilpotent subalgebra that consists of all the upper triangular matrices exceptthose in these blocks along the diagonal, so that slnd ∼= p⊕ n. The isomorphismgiven in Proposition 3.8 shows that sln(B) is isomorphic to p⊗C[t]⊕ n⊗ tC[t].

We gave in the previous subsection two triangular decompositions of sln(A).The Lie algebra slnd(C[t±1

]) admits similar decompositions, namely,

slnd(C[t±1])∼= (n−nd ⊗C[t±1

])⊕ (hnd ⊗C[t±1])⊕ (n+nd ⊗C[t±1

]),

slnd(C[t±1])∼= (slnd ⊗ t−1C[t−1

]⊕ n−nd)⊕ hnd ⊕ (n+

nd ⊕ slnd ⊗ tC[t]).

The isomorphism given in Proposition 3.8 preserves the second decomposition, butnot the first one.

We conclude that a lot is already known about the representation theory ofcyclic affine algebras, and simple finite-dimensional representations are classifiedby tuples of nd − 1 polynomials; see [Chari 1986]. We state this more explicitlyfor toroidal Lie algebras in Section 5A below.

4. Cyclic double affine Lie algebras

In this section, we set A=C[u±1, v±1]o0, B =C[u±1, v]o0, C =C[u, v]o0.

Here, ξ acts on u and v by ξ(u)= ζu and ξ(v)= ζ−1v. Note that, setting w= uv,we deduce that A is isomorphic to C[u±1, w±1

]o0, where 0 acts trivially on w.The same remark applies to B.

We will be interested in the structure of the Lie algebras sln(A), sln(B), sln(C)and their universal central extensions.

4A. Structure. We need to know certain cyclic homology groups. For example,

HC0(A)= A0, HC0(B)= B0, HC0(C)= C0⊕C⊕(|0|−1),

HC1(A) ∼= �1(A)0/d(A0), and similarly for B and C (see Corollary 4.3). Asvector spaces, when 0 6= {id}, we have

HC1(A)= vC[u±1, v±1]0du⊕C[u±1

]0v−1dv

= (C[u±1]0⊗C C[w±1

])u−1du⊕C[u±1]0w−1dw,

HC1(B)= vC[u±1, v]0du⊕Cu−1du∼= (wC[w]⊗C C[u±1

]0)u−1du⊕Cu−1du,

HC1(C)= vC[u, v]0du.


These results can be obtained from the computations of Hochschild homology in[Farinati 2005], which are valid for more general groups than 0. For A and B,they can also be deduced from the proof of this extension of Proposition 3.8:

Proposition 4.1. The Lie algebra sln(A) is isomorphic to the toroidal Lie algebraslnd(C[s±1, t±1

]); likewise sln(B) is isomorphic to slnd(C[s±1, t]).

Proof. We write A = C[u±1, w±1]. Since C[u±1

]o0 ∼= Md(C[s±1]) with s = ud

(see the proof of Proposition 3.8) and A∼= (C[u±]o0)⊗CC[w±1], we immediately

deduce the first claim by setting t = w. The same argument applies to B. �

Remark 4.2. The isomorphism in this proposition is reminiscent of [Berman et al.2003, Lemma 4.1]. In that article, the authors used vertex operator techniques toconstruct representations of a certain affinization of glN (C[G][t, t−1

]), where G isan admissible subgroup of C× (in the sense defined therein).

Explicitly, the isomorphism in Proposition 4.1 sends Eabuiw j ek with i =m+dl,−k ≤ m ≤ d − 1− k and 0 ≤ k ≤ d − 1 to Ea+(m+k)n,b+knsl t j ; in terms of u, vinstead of u, w, it maps Eabuiv j ek with i − j = m + dl and −k ≤ m ≤ d −1− k to Ea+(m+k)n,b+knsl t j . In particular, if i = 0, then this map restricts to theisomorphism of Proposition 3.8 for C[v±1

], with v playing the role of u−1. TheLie subalgebra sln(C[w

±1]o0) gets identified with the direct sum of d copies of

sln(C[t±1]), which agrees with C[w±1

]o0 ∼= C[w±1]⊗C C[0] ∼= C[w±1

]⊕d .

Corollary 4.3. The cyclic homology groups HC1(A) and HC1(B) are given by

HC1(A)∼=�1(C[s±1, t±1

])

d(C[s±1, t±1])and HC1(B)∼=

�1(C[s±1, t])d(C[s±1, t])

.

Proof. This follows from the algebra isomorphism A ∼= Md(C[s±1, t±1])) in the

proof of Proposition 4.1. (A similar isomorphism holds for B.) �

When we restrict the isomorphism of Proposition 4.1 to sln(C), we obtain aninjective map from sln(C) into slnd(C[s, t]). It comes from an injection of Cinto Md(C[s, t]). This latter map is a special case (n = 1 and x, y, ν = 0) ofthe homomorphism introduced in [Gordon 2007, Section 6.1]; see also [Crawley-Boevey 1991]. It is not an epimorphism from C to Md(C[s, t]) in the set theoreticalsense, but it is an epimorphism in the following categorical sense: Any two ringhomomorphisms Md(C[s, t]) →−→ D whose composites with C ↪→ Md(C[s, t])agree on C must be equal.

Using Proposition 4.1 and the explicit isomorphism given right after the proof,we can give a formula for the bracket on sln(A). It is easier to do it first with u, wand then translate the result to u, v. Note that we identify the center of sln(A) with�1(C[s±1, t±1

])/d(C[s±1, t±1]), so that it makes sense to write d( f ) for some


f ∈ C[s±1, t±1], with s = ud and t = w as above. For a commutative algebra R,

the bracket on sln(R) is given by

[Ea1b1r1, Ea2b2r2] = [Ea1b1, Ea2b2]r1r2+Tr(Ea1b1 Ea2b2)r1d(r2),

where Tr is the usual trace functional [Kassel 1984] and r1d(r2) ∈ �1(R)/d R.

Now consider

[Ea1+(m1+k1)n,b1+k1nsl1 t j1, Ea2+(m2+k2)n,b2+k2nsl2 t j2]

= δb1+k1n=a2+(m2+k2)n Ea1+(m1+k1)n,b2+k2nsl1+l2 t j1+ j2

− δb2+k2n=a1+(m1+k1)n Ea2+(m2+k2)n,b1+k1nsl1+l2 t j1+ j2

+ δa1+(m1+k1)n=b2+k2nδb1+k1n=a2+(m2+k2)nsl1 t j1d(sl2 t j2).

The isomorphism of Proposition 4.1 identifies the left side with

[Ea1b1ui1w j1ek1, Ea2b2ui2w j2ek2] where i1 = dl1+m1 and i2 = dl2+m2,

whereas the right side is identified with

δa2=b1δk1≡m2+k2 Ea1,b2ui1+i2w j1+ j2

− δb2=a1δm1+k1≡k2 Ea2b1ui1+i2w j1+ j2

+ δa1=b2δm1+k1≡k2δb1=a2δk1≡m2+k2ui1−m1w j1d(ui2−m2w j2).

Setting v = wu−1, we obtain a formula in terms of u and v if we now define m1

and m2 by ii − ji = mi + li d:

[Ea1b1ui1v j1ek1, Ea2b2ui2v j2ek2]

= δa2=b1δk1≡m2+k2 Ea1,b2ui1+i2v j1+ j2 − δb2=a1δm1+k1≡k2 Ea2b1ui1+i2v j1+ j2

+ δa1=b2δm1+k1≡k2δb1=a2δk1≡m2+k2ui1−m1v j1d(ui2−m2v j2).

The bracket when B and C replace A has exactly the same formula; simplyrestrict the values allowed for i1, i2, j1, j2. Proposition 2.3’s description of sln(A)implies that the natural maps sln(B), sln(C)→ sln(A) are embeddings.

In the next proposition, we use Proposition 4.1 to adapt [Moody et al. 1990,Proposition 3.5] to sln(A). Our Proposition 4.4 has a few more relations, but theproofs are the same.

Let C= (ci j ) be the affine Cartan matrix of type And−1 with rows and columnsindexed from 0 to nd − 1. Let f : [0, n − 1] × [0, d − 1] → [0, nd − 1] be thefunction f (i, j)= i + jn.

Proposition 4.4. The Lie algebra sln(A) is isomorphic to the Lie algebra t gener-ated by the elements X±i, j,r and Hi, j,r for 0 ≤ i ≤ n− 1, 0 ≤ j ≤ d − 1 and r ∈ Z,


and a central element c satisfying relations

[Hi1, j1,r1, Hi2, j2,r2] = r1c f (i1, j1), f (i2, j2)δr1+r2=0c,

[Hi1, j1,0, X±i2, j2,r2] = ±c f (i1, j1), f (i2, j2)X

±

i2, j2,r2,

[Hi1, j1,r1+1, X±i2, j2,r2] = [Hi1, j1,r1, X±i2, j2,r2+1],

[X±i1, j1,r1+1, X±i2, j2,r2] = [X±i1, j1,r1

, X±i2, j2,r2+1],

[X+i1, j1,r1, X−i2, j2,r2

] = δi1=i2δ j1= j2(Hi1, j1,r1+r2 + r1δr1+r2=0c),

ad(X±i1, j1,r1)1−c f (i1, j1), f (i2, j2)(X±i2, j2,r2

)= 0.

Remark 4.5. The elements with r = 0 generate a copy of sln(C[u±1]o0), and this

proposition gives a set of relations describing this algebra, which is the one in termsof Chevalley–Kac generators of slnd(C[t±1

])— see Proposition 3.8. The elementswith i 6= 0 generate a central extension of sln(C[w

±1])⊕d , and this proposition

gives a presentation of sln(C[w±1]), which is the one obtained by considering a

Cartan matrix of finite type An−1 in [Moody et al. 1990, Proposition 3.5].

An isomorphism τ : t ∼−→ sln(A) is given explicitly on the generators by thefollowing formulas

(X+i, j,r , X−i, j,r ) 7→

(Ei,i+1w

r e j , Ei+1,iwr e j ) if i 6= 0,

(En1u−1wr e j , E1nuwr e j−1) if i = 0, j 6= 0,(En1u2d−1wr ed−1, E1n ⊗ u−(2d−1)wr e0) if i = j = 0,

Hi, j,r 7→

(Ei i − Ei+1,i+1)w

r e j if i 6= 0,Ennw

r e j−1− E11wr e j if i = 0, j 6= 0,

Ennwr e0− E11w

r ed−1− dwr u−1du if i = j = 0,

c 7→ w−1dw = u−1du+ v−1dv.

It is possible to obtain similar presentations for sln(B) and sln(C), which areLie subalgebras of sln(A).

4B. Derivations in the toroidal case. The Kac–Moody Lie algebras of affine typeassociated to the semisimple Lie algebra g are obtained by adding a derivation to theuniversal central extension of g⊗C C[t±1

]. People who work on extended affineLie algebras know to extend sln(C[s±1, t±1

]) by adding derivations. Following[Allison et al. 1997], we interpret this in the context of sln(A), slm(B), sln(C).

To sln(A), we add derivations du and dvw that satisfy the commutation relations

[du,m⊗ ukwlγ] = 0 if k 6≡ 0 mod d,

[du,m⊗ udkv jγ] = km⊗ udkw jγ,

[dvw,m⊗ ukwlγ] = lm⊗ ukwlγ.


We observe that

[dvw,m⊗ ukvlγ] = lm⊗ ukvlγ and [du,m⊗ ukvlγ] = δk≡lk−l

dm⊗ ukvlγ.

We could define similarly dv and duw. We then have the relation du = −dv and

d · du = duw − dvw. Dropping the index w, we can add two derivations du and dv

to sln(A).

Definition 4.6. The cyclic double affine Lie algebra sln(A) is defined by addingthe derivations du and dv to sln(A). We define similarly sln(B) and sln(C).

4C. Triangular decompositions. We have the triangular decompositions

(1) sln(A)∼= (n−A)⊕ (hA⊕ I [A, A])⊕ (n+A),

and sln(A) is also isomorphic to

(2) slnu−1C[u−1, v±1]o0⊕

( d−1⊕i=1

j≤−1

I (C[v±1]u jξ i )⊕

⊕s≤−1r 6≡s

I (Cusvr )

)

⊕n−C[v±1]o0⊕

(hC[v±1

]o0⊕( ⊕

1≤i≤d−1

I (C[v±1]ξ i )⊕

⊕r 6≡0

I (Cvr )

))

⊕slnuC[u, v±1]o0⊕

(d−1⊕i=1j≥1

I (C[v±1]u jξ i )⊕

⊕s≥1r 6≡s

I (Cusvr )

)⊕n+C[v±1

]o0.

and then by, setting w = uv, we see that sln(A) is also isomorphic to

(3) slnu−1C[u−1, w±1]o0⊕

( d−1⊕i=1

j≤−1, r∈Z

I (Cwr u jξ i )⊕⊕s≤−1

s 6≡0, r∈Z

I (Cwr us)

)

⊕n−C[w±1]o0⊕

(hC[w±1

]o0⊕( ⊕

1≤i≤d−1

I (C[w±1]ξ i )

))⊕slnuC[u, w±1

]o0

⊕

( d−1⊕i=1

j≥1,r∈Z

I (Cwr u jξ i )⊕⊕s≥1

s 6≡0,r∈Z

I (Cwr us)

)⊕ n+C[w±1

]o0.

In the last two decompositions, one can exchange u and v and get other decompo-sitions.

The universal central extensions sln(A), sln(B) and sln(C) also have three tri-angular decompositions They are obtained by adding the center to the middle part.

It is worth looking quickly at hA⊕I [A, A]. We know that A∼=Md(C[s±1, t±1]),

so, if d > 1, then [A, A] = A and hA⊕ I [A, A] ∼= gld(C[s±1, t±1])⊕n .


5. Representations of cyclic double affine Lie algebras

In this section, we begin to study the representation theory of the algebras definedin the previous sections.

5A. Integrable and highest weight modules for cyclic double affine Lie algebras.We have just presented three triangular decompositions of sln(A). The first one isanalogous to the one used for Weyl modules in [Feigin and Loktev 2004], but inour situation the middle Lie algebra is not commutative if 0 6= {id}. (Note that ananalogue of the first triangular decomposition is not known for quantum toroidalalgebras.) We will return to Weyl modules in Section 5B below.

The second triangular decomposition corresponds to the first triangular decom-position of the cyclic affine Lie algebras for the parameter v, and to the secondof the cyclic affine Lie algebra for the parameter u. Thus it is analogous to thetriangular decomposition used in [Miki 2000; Nakajima 2001; Hernandez 2005]to construct l-highest weight representations of quantum toroidal algebras. Again,in our situation, the middle Lie algebra is not commutative if 0 6= {id}.

However, the middle term H of the last triangular decomposition is a commu-tative Lie algebra. Actually, this last case is obtained by considering the firsttriangular decomposition of the cyclic affine Lie algebras for the parameter wand the second triangular decomposition of the cyclic affine Lie algebra for theparameter u. Under the isomorphism between sln(A) and the toroidal Lie algebraslnd(C[u±1, w±1

]) given in Proposition 4.1, it corresponds to the standard decom-position of slnd(C[u±1, w±1

]) as used in [Chari and Le 2003] for a certain centralextension of slnd(C[u±1, w±1

]). (It is analogous to the decomposition consideredin the quantum case [Miki 2000; Nakajima 2001; Hernandez 2005]). In particular,the notions of integrable and highest weight modules for this decomposition havebeen studied in [Chari and Le 2003; Rao 2004; Yoon 2002]; we simply reformulatetheir results for the benefit of the reader in the next subsection. Integrable highestweight representations are classified by tuples of nd−1 polynomials. In oppositionto the quantum case, evaluation morphisms are available and provide a direct wayto construct integrable representations.

The standard highest weight structure on sln(A) and sln(B). In this subsection,we include previously known results about the standard highest weight structureon sln(A) and on sln(B). Actually, the results below have been proved only forsln(A), but the proofs are similar for sln(B). Let g± be the positive and negativeparts of the triangular decomposition (3), and let H be the middle part.

Instead of highest weight vectors, we have to consider the notion of pseudo-highest weight vectors. Suppose that

3= (λi, j,r )0≤ j≤d−10≤i≤n−1, r∈Z, where λi, j,r ∈ C.


We define the Verma module M(3) to be the sln(A)-module induced from theH⊕ g+ representation generated by the vector v3 on which g+ and c act by zeroand Hi, j,r acts by multiplication by λi, j,r (in the notation of Proposition 4.4). Wehave a grading on the Verma module and so it has a unique simple quotient L(λ).For µ∈H∗, we define the notion of weight space Vµ of a representation V as usual.

Definition 5.1. A module M over sln(A) is integrable if M =⊕

µ∈H∗ Mµ and ifthe vectors Ei j usvrγ act locally nilpotently if 1≤ i 6= j ≤ n, r, s ∈ Z and γ ∈ 0.

Note that, in the quantum case, a stronger notion of integrability is used insteadof local nilpotency of the operators [Chari and Le 2003; Rao 2004].

Proposition 5.2 [Chari and Le 2003]. The irreducible module L(3) is integrableif and only if , for any 0 ≤ i ≤ n− 1 and 0 ≤ j ≤ d − 1, we have λi, j,0 ∈ Z≥0 andthere exist monic polynomials Pi, j (z) of degree λi, j,0 such that

∑r≥1

λi, j,r zr−1= −

P ′i, j (z)

Pi, j (z)and

∑r≥1

λi, j,−r zr−1=−λi, j,0z−1

+ z−2P ′i, j (z

−1)

Pi, j (z−1)

as formal power series.

The proof of the necessary condition in this proposition reduces to the case ofthe loop Lie algebra of sl2, which is why it extends automatically from the affine tothe double affine setup. The sufficiency is proved as in [Chari and Le 2003] usingtensor products of evaluation modules (the formulas for the power series are a bitdifferent from those in [Chari and Le 2003, Proposition 3.1] as we use differencevariables; see also [Miki 2004]). In the quantum context, the polynomials Pi, j (z)are called Drinfeld polynomials. A similar criterion for integrability exists forquantum toroidal algebras; this is explained in [Hernandez 2005] after proving thatcertain subalgebras of a quantum toroidal algebras are isomorphic to the quantizedenveloping algebra of sl2(C[t±1

]). Affine Yangians for sln are built from copies ofY (sl2), the Yangian for sl2: this follows from the PBW property of affine Yangiansproved in [Guay 2007]; hence a similar integrability condition holds for them also.

The standard highest weight structure on sln(C). The case that interests us moreand presents some novelty is sln(C), because the Lie algebra sln(C) is not isomor-phic to slnd(C[s, t]). The three triangular decompositions of sln(A) given at theend of the Section 4 yield such decompositions for sln(C), and we consider theone coming from (3). More precisely, sln(C) can be decomposed into the direct


sum of three subalgebras, as

(4)(

slnvC[v,w]o0⊕( ⊕

j≥11≤i≤d−1

I (C[w]v jξ i )⊕⊕r≥1r 6≡0

I (C[w]vr )

)⊕n−C[w]o0

)

⊕

(hC[w]o0⊕

( ⊕1≤i≤d−1

I (wC[w]ξ i )

))

⊕

(slnuC[u, w]o0⊕

( ⊕j≥1

1≤i≤d−1

I (C[w]u jξ i )⊕⊕r≥1r 6≡0

I (C[w]ur )

)⊕n+C[w]o0

).

We have an embedding sln(C) ↪→ slnd(C[u±1, v]), but in order to classify inte-grable, highest weight representations of sln(C), we will instead use the followingpresentation.

Proposition 5.3. The Lie algebra sln(C) is isomorphic to the Lie algebra k that isgenerated by the elements X±i, j,r , Hi, j,r , X+0, j,r , X−0, j,r+1, H0, j,r+1 for 1≤ i ≤ n−1,0≤ j ≤ d − 1 and r ∈ Z≥0, and that satisfies the relations

[Hi1, j1,r1, Hi2, j2,r2] = 0,

[Hi1, j1,0, X±i2, j2,r2] = ±c f (i1, j1), f (i2, j2)X

±

i2, j2,r2,

[Hi1, j1,r1+1, X±i2, j2,r2] = [Hi1, j1,r1, X±i2, j2,r2+1],

[X±i1, j1,r1+1, X±i2, j2,r2] = [X±i1, j1,r1

, X±i2, j2,r2+1],

[X+i1, j1,r1, X−i2, j2,r2

] = δi1=i2δ j1= j2 Hi1, j1,r1+r2,

ad(X±i1, j1,r1)1−c f (i1, j1), f (i2, j2)(X±i2, j2,r2

)= 0.

The proof of Proposition 4.4 in [Moody et al. 1990] works also for sln(B), andit is possible to deduce from it Proposition 5.3. See [Guay 2009b] for more details.

The elements with i 6= 0 generate a Lie subalgebra isomorphic to sln(C[w]o0),the elements X+i, j,r for 0≤ i ≤n−1, 0≤ j ≤d−1 and r ∈Z≥0 generate the positivepart sln(C)+ of the decomposition (4), and, finally, the elements X−i, j,r and X−0, j,r+1for 0≤ i ≤ n−1, 0≤ j ≤ d−1 and r ∈Z≥0 generate the decomposition’s negativepart sln(C)−. Note that sln(C)+ ∼= sln(C)− via X+i, j,r 7→ X−i, j,r for 1 ≤ i ≤ n− 1and X+0, j,r 7→ X−0, j,r+1. The elements with r = 0 generate a copy of sln(C[u]o0),whereas the elements X±i, j,0 and X−0, j,1 with 1 ≤ i ≤ n − 1 and 0 ≤ j ≤ d − 1generate a Lie subalgebra isomorphic to sln(C[v]o0). For a fixed 0≤ j ≤ d−1,the elements X+0, j,r , X−0, j,r+1, H0, j,r+1 for all r ∈ Z≥0 generate a subalgebra ofsl2(C[w]) which, as a vector space, is n−2 wC[w]⊕h2wC[w]⊕n+2 C[w], where thesubscript 2 labels the corresponding subalgebra of sl2. We denote this subalgebraof sl2(C[w]) by ˇsl2(C[w]).


Integrability of representations of sln(C) has the same meaning as it did inDefinition 5.1. As for sln(A) and sln(B), we have Verma modules M(3) andtheir irreducible quotients L(3) for each pseudoweight 3 = (λi, j,r ∈ C) with0≤ i ≤ n−1, r ∈ Z and 0≤ j ≤ d−1 but r ≥ 1 if i = 0; the highest weight cyclicgenerator is again denoted v3.

Proposition 5.4. The irreducible module L(3) is integrable if and only if λi, j,0

belongs to Z≥0 for any 1 ≤ i ≤ n − 1 and 0 ≤ j ≤ d − 1, and there exist monicpolynomials Pi, j (z) for the same range of i and j such that

∑r≥1 λi, j,r z−r−1 is

equal to −deg(Pi, j ) + P ′i, j (z)/Pi, j (z) as formal power series and Pi, j (z) is ofdegree λi, j,0 if 1≤ i ≤ n− 1 and 0≤ j ≤ d − 1.

Let us say a few words about the proof. The main difference with the casessln(A) and sln(B) is that we do not have the elements X−0, j,0 for 0 ≤ j ≤ d − 1.However, it is still possible to apply argument of Chari [1986] in the affine sl2-case, modulo some small differences. For instance, [Chari 1986, Proposition 1.1]is fundamental for the rest of that article, but it cannot be applied in our casewhen i = 0. What we need instead is an expression for (X0, j,0)

r (X−0, j,1)r . But

Chari’s proposition is a consequence of [Garland 1978, Lemma 7.5]. To obtainan expression for (X0, j,0)

r (X−0, j,1)r , we just have to apply the automorphism of

ˇsl2(C[w]) given by

E21wr+17→ E12w

r , E12wr7→ E21tr+1, (E11− E22)tr+1

7→ (E22− E11)tr+1

for r ∈ Z≥0. We lose the condition that the degree of P0, j (z) is λ0, j (here thedegree of P0, j (z) is the smallest integer r such that (X−0, j,1)

r+1v3 = 0) becausesln(C) does not contain the sl2-copies generated by X±0, j,0 and H0, j,0. The proof ofthe sufficiency of the condition in the proposition consists in the construction of anintegrable quotient of the Verma module M(3) using tensor products of evaluationmodules, as in [Chari and Le 2003]. Note that also in the case of sln(C), the degreeof P0, j (z) is the smallest integer r such that (X−0, j,1)

r+1 acts by zero on the cyclichighest weight vector.

5B. Weyl modules for sln(C[u, v]o0). For a Lie algebra g and a commutative C-algebra A, we may denote g⊗CA by g(A) or gA. If g=n+⊕H⊕n−, the Weyl mod-ules [Feigin and Loktev 2004] are certain representations of g(A) generated by aweight vector v satisfying (n+(A)).v= 0. (In this subsection, we consider only thelocal Weyl modules, not the global ones.) The motivation to study Weyl modulesis that they should be simpler to understand than the finite-dimensional irreduciblemodules. This is what happens in the quantum affine setup where the Weyl mod-ules for the affine Lie algebras are closely related to finite-dimensional irreduciblemodules of the corresponding affine quantum group when q 7→ 1; see [Chari andPressley 2001; Chari and Loktev 2006]. The definition of Weyl modules depends


on the choice of a triangular decomposition, but only the first of our triangulardecompositions for cyclic double affine Lie algebras seems appropriate. It shouldbe noted that we cannot use Proposition 4.1 to deduce results about Weyl modulesfor sln(A) in our context because, when 0 is nontrivial, the isomorphism in thatproposition does not map the triangular decomposition (1) of sln(A) to the decom-position slnd(C[s±1, t±1

]) = n−ndC[s±1, t±1] ⊕ hndC[s±1, t±1

] ⊕ n+ndC[s±1, t±1]

considered in [Feigin and Loktev 2004].In this subsection, we need a definition of integrability stronger than the one

used in Definition 5.1, namely, the one used in [Feigin and Loktev 2004].

Definition 5.5. A module M over a Lie algebra of the type g⊗C A is said to beintegrable if Mµ is nonzero for only finitely many µ ∈ P .

When A is the coordinate ring of an affine algebraic variety X , Weyl modulesare associated to multisets of points of X . In the simplest case of a (closed) point,we have an augmentation A→ C. However, when it comes to the triangular de-composition (1), the middle term is isomorphic to d⊗C Md(C[s±1, t±1

]), whered is the abelian Lie algebra of the diagonal matrices in gln . When d = 1, weare exactly in the same situation as in [Feigin and Loktev 2004] (with X the two-dimensional torus C××C×), but when d > 1, the Lie algebra is noncommutative.One new possibility is to consider maximal two-sided ideals in A o0 or, equiva-lently, augmentation maps. We are thus led to the following definition, which wecan formulate in a more general setting.

Definition 5.6. Let A be a commutative, finitely generated algebra with a unit,and let G be a finite group acting on A by algebra automorphisms. Consider anaugmentation ε of A o G, that is, an algebra homomorphism A o G→ C, and letλ ∈ h∗ be a dominant integral weight. We define the Weyl module W ε

AoG to be themaximal integrable cyclic sln(A o G)-module generated by a vector vλ such that,for a ∈A o G,

(ha)(vλ)= λ(h)ε(a)vλ and n+(A o G)(vλ)= 0.

The existence of such a maximal module can be proved as in [Feigin and Loktev2004] using the notion of global Weyl module. This definition agrees with the oneused in that paper in the case G = {id}. We note that, by sl2-theory, we have thatf λ(hi )+1i vλ = 0, where, as usual, we denote by fi , hi , ei for 1 ≤ i ≤ n − 1 the

standard Chevalley generators of sln .It turns out that Weyl modules for the smash product A o G are related to

Weyl modules for a much smaller ring. We have a decomposition of G-modulesA=AG

⊕A′, where A′ is the subrepresentation without invariants. Let us denoteby A the quotient of A by the two-sided ideal generated by A′. Note that it may be


much smaller than AG — it can even reduce to C, for instance, when G is Z/dZ

and A= C[u] or even A= C[u, v].Consider an augmentation ε of A o G. Note that A′ ⊂ [A o G,A o G], so

ε(A′)= 0 and ε descends to an augmentation ε of A.

Theorem 5.7. Let λ ∈ h∗ be a dominant integral weight. We have an isomorphismof modules over sln(A o G) given by

W εAoC[G](λ)

∼=W εA(λ).

We have a surjective map AoG→A; hence sln(AoG) acts on W εA(λ) and this

yields a surjective map W εAoG(λ)→W ε

A(λ) of modules over sln(AoG). The ringA is the quotient of A o G by the ideal Jε generated by A′ and elements γ− ε(γ)in C[G] for γ ∈ G. We need only show that sln ⊗ Jε acts on W ε

AoG(λ) by zero.Actually, since sln ⊗ Jε is an ideal, it is enough to show that it acts by zero on

the highest weight vector vλ. As ε(Jε)= 0, this is true for b⊗ Jε , so it remains toprove our claim for Ei j ⊗ ε with i > j . Now the question is reduced to sl2-case,so, to simplify the notation, let us set f = Ei j , e = E j i and h = Ei i − E j j .

Lemma 5.8. Let P ∈A o0. If ε(P)= 0, then ( f ⊗ Pλ(h))vλ = 0.

Proof. We already know that f λ(h)+1vλ= 0. Applying e⊗ P a total of j times andusing the assumption ε(P)= 0 yields ( f λ(h)+1− j

⊗ P j )vλ = 0, so taking j = λ(h)proves the lemma. �

Lemma 5.9. If ( f ⊗ P)vλ = 0 and Q ∈ A, then ( f ⊗ (P Q + Q P))vλ = 0.Also, if Q belongs to the commutator [sln(A o G), sln(A o G)], then we have( f ⊗ P Q)vλ = ( f ⊗ Q P)vλ = 0.

Proof. Applying f ⊗ P to both sides of (h⊗ Q)vλ = λ(h)ε(Q)vλ yields the firstequality. If now, say, Q= Q1 Q2−Q2 Q1, then (Ei i+E j j )⊗Q=[h⊗Q1, h⊗Q2];starting from ( f ⊗ Q)vλ = 0 and applying Ei i ⊗ Q or E j j ⊗ Q to both sides, weobtain the second equality. (Note here that Ei i ⊗ Q belongs to sl2(A o G) since,by assumption, Q ∈ [sln(A o G), sln(A o G)].) �

Proof of Theorem 5.7. First let us show that

(5) ( f ⊗ (x − ε(x)))vλ = 0 for x ∈ C[G].

Note that C[G]=Ccε⊕ Iε , where c2ε = cε , ε(cε)= 1, and Iε is the kernel of ε|C[G].

We have I l+1ε = Iε , so Lemma 5.8 yields Equation (5) for x ∈ Iε . Also note that

cε − 1 belongs to Iε , so we have (5) also for x = cε and, therefore, for any x .Since A′ ∈ [sln(AoG), sln(AoG)] and AG commutes with C[G], Lemma 5.9

implies that ( f ⊗A(x − ε(x)))vλ = 0.


It remains to show that ( f ⊗A′)vλ = 0. By Lemma 5.9, for any a ∈ A′ andγ ∈ G, we have

( f ⊗ γa)vλ = ( f ⊗ aγ)vλ = ( f ⊗ aε(γ))vλ.

So( f ⊗ (a− γ−1(a)))vλ =

1ε(γ)

( f ⊗ (γa− a))vλ = 0.

Finally, note that the elements (a− γ−1(a)) for a ∈A′ and γ ∈ G span A′. �

5C. Weyl modules associated to rings of invariants. When A is a commutative,unital, finitely generated C-algebra, Weyl modules for sln ⊗C A can be attachedto multisets of points in Spec(A) or, more generally, to any ideal in A. In thissubsection, we first apply an approach due to Feigin and Loktev [2004] and Chariand Pressley [2001] to describe certain local Weyl modules for sln(C[u, v]) andsln(C[u, v]0). This approach realizes them as the Schur–Weyl duals of certainmodules of coinvariants. A natural question to ask is, What is the dimension ofthese local Weyl modules? For loop algebras sln(C[u, u−1

]), this question wasfully answered in [Chari and Loktev 2006]. For sln(C[u, v]) and a multiple of thefundamental weight of the natural representation of sln on Cn , this problem wassolved in [Feigin and Loktev 2004], but the answer relies on the difficult theoremof M. Haiman [2002] on the dimension of diagonal harmonics. To compute thedimension of the Weyl modules that we introduce below, we would need an ex-tension of Haiman’s theorem to certain rings of coinvariants attached to wreathproducts of the cyclic group Z/dZ, but this is still an open problem as far as weknow. At least, we are able to provide a lower bound for the dimension of somelocal Weyl modules by using a theorem of R. Vale [2007], which generalizes anearlier result of I. Gordon [2003].

Definition 5.10. Let U be a representation of sln⊗A and let µ∈ h∗. Suppose thatwe have an augmentation map ε : A→ C. A vector vµ ∈ U is called a highestweight vector if (g⊗a)vµ = 0 when g ∈ n+, a ∈A and (h⊗a)vµ = µ(h)ε(a)vµfor all h ∈ h and a ∈A.

Theorem 5.11 [Feigin and Loktev 2004]. Let µ ∈ h∗ be a dominant integralweight. There exists a universal finite-dimensional module W A

ε (µ) such that anyfinite-dimensional representation of sln ⊗A generated by a highest weight vectorvµ is a quotient of W A

ε (µ).

Recall a general result of [Feigin and Loktev 2004]; see also [Loktev 2008]. Thesymmetric group Sl acts on A⊗l , so we can form DHl(A)=A⊗l/(Syml(A)ε), thequotient of A⊗l by the ideal generated by the tensors that are invariant under theaction of Sl and that are in the kernel of ε (extended as an augmentation Sym(A)).When A= C[u, v], this quotient is called the space of diagonal coinvariants.


If E is a representation of Sl , denote by SWnl (E) the representation of sln given

by ((Cn)⊗l⊗E)Sl . This is the classical Schur–Weyl construction. Note that DHl(A)

is a representation of Sl . More generally, as is observed in [Loktev 2008], if E isa representation of the smash product (A⊗l)o Sl , then SWn

l (E) is a representationof sln(A).

Let ω1 be the fundamental weight of sln that is the highest weight of its naturalrepresentation Cn .

Theorem 5.12 [Feigin and Loktev 2004]. The Weyl module W Aε (lω1) of sln(A) is

isomorphic to SWnl (DHl(A)).

When A = C[u, v], the dimension of the ring of diagonal harmonics DHl(A)

is a difficult result proved by M. Haiman [2002]. Until after Proposition 5.15,we will assume that 0 is an arbitrary finite subgroup of SL2(C). The precedingtheorem can be applied to A=C[u, v]G and ε :C[u, v]G→C, the homomorphismgiven by the maximal ideal C[u, v]G

+corresponding to the singularity. (Here, G

is an arbitrary finite subgroup of SL2(C).) It gives a nice description of the Weylmodule for multiples of ω1, but, to compute its dimension, we would have to knowmore about the structure of DHl(C[u, v]G) as a module for Sl . As far as we know,this is still an open problem when G 6= {1}. We are, however, able to obtain a partialresult by considering the ring A = C[u, v] but with highest weight conditions onh⊗C[u, v]0, and from it we can deduce a lower bound when A= C[u, v]0.

Denote by HG,l the quotient of C[u1, . . . , ul, v1, . . . , vl] by the ideal generatedby Sl n G×l-invariants with zero at the origin. This is a module for Sl n G×l andalso for (Sl n G×l)n C[u, v]⊗l .

Definition 5.13. Let µ ∈ h∗ be a dominant integral weight, and let WG(µ) bethe maximal finite-dimensional module over sln(C[u, v]) generated by a vector vµsuch that (n+⊗C[u, v])vµ = 0 and

(h⊗ P)vµ = µ(h)P(0, 0)vµ for h ∈ h and P ∈ C[u, v]G .

We say that WG(µ) is the Weyl module for sln(C[u, v]) associated to the ideal(C[u, v]G

+).

Remark 5.14. The existence of a maximal finite-dimensional module with thisproperty can be proved as in [Feigin and Loktev 2004].

Proposition 5.15. The Weyl module WG(lω1) is Schur–Weyl dual to HG,l , that is,WG(lω1)= SWn

l (HG,l).

Proof. The argument is the same as the one used in [Feigin and Loktev 2004],so we just sketch it. The Weyl module WG(lω1) is the quotient of the globalWeyl module of sln(C[u, v]) for the weight lω1 by the submodule generated by(h⊗C[u, v]G

+)vlω1 . The global Weyl module for this weight is Syml(Cn

⊗C[u, v]),


and we must quotient by the submodule generated by the action of Syml(C[u, v]G+).

Thus the Weyl module WG(lω1) is obtained by applying the Schur–Weyl con-struction to the quotient of C[u1, . . . , ul, v1, . . . , vl] by the ideal generated by theSl n G×l-invariant polynomials. �

The following theorem of R. Vale is a generalization of a theorem of I. Gordon[2003] for Sl .

Theorem 5.16 [Vale 2007]. The representation H0,l has a quotient H 00,l such that

the trace on H 00,l ⊗ Sign of a permutation σ ∈ Sl consisting of s cycles is equal to

(dl + 1)s .

Now let us apply Theorem 5.16 to the character calculation for W0(lω1). LetF(l, k) be the set of functions from {1, . . . , l} to {1, . . . , k}. This set admits an ac-tion of Sl by permutation of the arguments. Denote by CF(l, k) the correspondingcomplex representation of Sl .

Lemma 5.17. Suppose that σ ∈ Sl is a product of s cycles. Then the trace of σ onCF(l, k) is equal to ks .

Proof. The trace of σ is equal to the number of functions stable under the actionof σ . A function is stable if it has the same value on all the elements of each cycle,so it is determined by s elements of {1, . . . , k}. �

Lemma 5.18. The sln-module SWnl (CF(l, k)⊗Sign) is isomorphic to

∧l((Cn)⊕k).

Proof. Note that (Cn)⊗l⊗ CF(l, k) is isomorphic to ((Cn)⊕k)⊗l as an SLn ×Sl-

module. The isomorphism can be constructed as the map sending (v1⊗ · · ·⊗ vl)⊗

f to v( f (1))1 ⊗· · ·⊗v

( f (l))l , where v(i) belongs to the i-th summand of (Cn)⊕k . Then

the lemma follows by restricting this isomorphism to the Sign component. �

Theorem 5.19. The Weyl module W0(lω1) has a quotient that, as a representationof sln , is isomorphic to

∧l((Cn)⊕(dl+1)).

Proof. Since H 00,l is a quotient of H0,l , we have by Proposition 5.15 that SWn

l (H00,l)

is a quotient of W0(lω1). Then Lemma 5.17 and Theorem 5.16 imply that H 00,l is

isomorphic to CF(l, k)⊗ Sign with k = dl + 1. By Lemma 5.18, SWnl (H

00,l) is

thus equal to∧l((Cn)⊕(dl+1)). �

Corollary 5.20. The dimension of W0(lω1) is bounded below by(n(dl+1)

l

).

By modifying slightly the argument in the previous paragraphs, we can givea lower bound also for some local Weyl modules when A = C[u, v]0. Let usintroduce an action of 0×l on CF(l, k): If we fix a generator ξi of the i-th copy of0 in 0×l and if f ∈ CF(l, k), then ξi ( f )= ζ j−1 f if f (i)= j . This action can becombined with the one associated to Sl to obtain an action of 0×l oSl . The trace ofξi on CF(l, k) is

(∑k−1j=0 ζ

j)kl−1, which equals

(∑kj=0 ζ

j)kl−1, where 0≤k≤d−1

and k− 1≡ k mod (d). In particular, if k ≡ 1 mod (d), then this trace is kl−1.


Theorem 5.16 (see [Vale 2007]) also states that the trace of ξi on H 00,l⊗Sign is

equal to (dl+1)l−1. That theorem actually applies to any element in 0×l o Sl , andfrom it we can deduce that we have an isomorphism of 0×l o Sl-modules betweenH 00,l and CF(l, ld + 1)⊗Sign. (0×l acts trivially on Sign.) Therefore,

(H 00,l)

0×l∼= CF(l, l + 1)⊗Sign

since the functions in CF(l, ld+1) that are invariants under 0×l can be identifiedwith CF(l, l + 1).

Now, we can repeat the argument we used above. SWnl ((H

00,l)

0×l) is a quotient

of SWnl (H

0×l0,l ) isomorphic to

∧l((Cn)⊕(l+1)). Let W0(µ) be defined as W0(µ) inDefinition 5.13, but with C[u, v] replaced by C[u, v]0. The Weyl module W0(lω1)

is isomorphic to SWnl (DHl(C[u, v]0)). Since DHl(C[u, v]0)∼= (H0,l)0

×l, we con-

clude that W0(lω1) has a quotient isomorphic to∧l((Cn)⊕(l+1)) as an sln-module,

whence the following corollary.

Corollary 5.21. The dimension of W0(lω1) is bounded below by(n(l+1)

l

).

There is no reason to expect that the lower bound in Corollary 5.20 is the bestpossible. Indeed, we can show that it is too low when d = 2 and l = 4 by followingideas of I. Gordon. In this case, 0×l o Sl is isomorphic to the Weyl group W oftype B4. M. Haiman [1994] explains that the ring of diagonal coinvariants in thiscase has dimension 94

+ 1, which is one more than the dimension of a certainquotient introduced in [Haiman 1994, Conjectures 7.1.2, 7.1.3 and 7.2.3]. Theseconjectures were proved by I. Gordon [2003] and we denoted above this quotientby H 0

0,l .This means that, in HZ/2Z,4, there is a one-dimensional subspace E that carries a

nontrivial representation of W . There is an action of sl2(C) on HZ/2Z,4 commutingwith the action of S4 (this is actually true in C[u1, v1, . . . , ul, vl] for any l ∈ Z≥1),so E is also a representation of sl2(C) and must thus be trivial. The standarddiagonal element h ∈ sl2(C) acts by

∑4i=1(ui d/dui − vi d/dvi ), so this operator

acts trivially on E , which implies that the monomials that appear in E have theiru-degree equal to their v-degree.

The Weyl module WZ/2Z(4ω1) is obtained by applying the Schur–Weyl functorto HZ/2Z,4, so, if n ≥ 4, WZ/2Z(4ω1) has dimension greater than

(9n4

), which shows

that the lower bound in Corollary 5.20 is too low.

6. Matrix Lie algebras over rational Cherednik algebras of rank one

The polynomial ring C[u, v] can be deformed into the first Weyl algebra A1 =

C〈u, v〉/(vu − uv − 1), which itself can be viewed as the ring D(C) of algebraicdifferential operators on the affine line A1

C. Such differential operators play an im-

portant role in the representation theory of Cherednik algebras, and the G-DDCA


of [Guay 2005; 2007; 2009b] are also deformations of the enveloping algebra ofgln(A1 o0) when G is a finite subgroup of SL2(C).

More generally, the rational Cherednik algebra Ht,c(Gl) for the wreath productGl = G×l o Sl admits two specializations of particular interest:

Ht=0,c=0(Gl)∼= C[x1, y1, . . . , xl, yl]o Gl and Ht=1,c=0(Gl)∼= Al o Gl,

where Al is the l-th Weyl algebra. The representation theories of these two algebrasdiffer greatly. For instance, in the first case, Ht=0,c=0(Gl) has infinitely many ir-reducible finite-dimensional representations, whereas Ht=1,c=0(Gl) has none. Ac-tually, Ht=1,c(Gl) does not have any finite-dimensional representations for genericvalues of c. The 0-DDCA also admit two such specializations, and it is reasonableto expect that their representation theories will thus differ noticeably. In this article,we want to start investigating the categories of modules for these two specializa-tions, so, in this section we will study matrix Lie algebras over rational Cherednikalgebras of rank one with t 6= 0.

Definition 6.1. Let c = (c1, . . . , cd−1) ∈ Cd−1. The rational Cherednik algebraHt,c(0) of rank one is the algebra generated by elements u, v, γ with γ ∈0=Z/dZ

and the relations γuγ−1= ζu, γvγ−1

= ζ−1v and

(6) vu− uv = t +d−1∑i=1

ciξi , where ξ is a generator of 0.

It will be convenient to rewrite (6) in the form vu−uv= t+∑d−1

i=0 ci (ei−ei+1)

for some ci ∈ C. We will need to use later the element ω that can be written in thethree equivalent ways

ω =−uv+d−1∑i=0

ci ei+1 =−vu+ t +d−1∑i=0

ci ei

=−uv+vu

2+

t2+

12

d−1∑i=0

ci (ei + ei+1).

Then one can check that [ω, u] = −tu and [ω, v] = tv.

Definition 6.2. Let c = (c1, . . . , cd−1) ∈ Cd−1. We will call the trigonometricCherednik algebra of rank one the algebra Ht,c(0)= C[u±1

]⊗C[u] Ht,c(0).

Remark 6.3. Trigonometric Cherednik algebras exist only for Weyl groups (realCoxeter groups), but we propose to use the terminology in the previous definitionbecause it is convenient. Moreover, as is explained in [Guay 2009b], Ht,c(0) de-pends actually only on the t parameter, that is, Ht,c(0)∼=Ht,c=0(0), but this is not


true for Ht,c(0). An explicit isomorphism Ht,c(0) ∼−→Ht,c=0(0) is given by

v 7→ v+

( d−1∑i=1

ci

1− ζ−i ξi)

u−1.

Note also that Ht,c(0) is generated by ω, u, u−1, 0 and that [ω, u−1] = tu−1.

The associative algebras Ht,c(0) and Ht,c(0) can be turned into Lie algebrasin the usual way, and the representation theory of a central extension of the Liealgebra gln(A1)was studied in [Boyallian et al. 1998; Kac and Radul 1993]. (Here,A1 is the algebra of differential operators on C×.) Boyallian and Liberati [2002]considered the case of the quantum torus Dq(C

×) = C〈u±1, v±1〉/(vu = quv).

In this section, we present some results about the structure of the Lie algebrassln(Ht,c(0)) and sln(Ht,c(0)), mostly when t 6= 0.

The (Lie) algebras Ht,c(0) and Ht,c(0) are graded as deg(u)=−1, deg(v)= 1and deg(γ) = 0. This induces gradings on the associative algebras Mn(Ht,c(0))

and Mn(Ht,c(0)) and on the Lie algebras gln(Ht,c(0)) and gln(Ht,c(0)). However,we will consider instead the grading

deg(Ei jvr usγ)= (r − s)n+ j − i.

In the case Ht=1,c=0(0 = {1}), this is the opposite of the principal Z-gradationconsidered in [Boyallian et al. 1998]. The graded pieces of degree k will be denotedgln(Ht,c(0))[k] and gln(Ht,c(0))[k].

6A. Central extensions. It was found in [Alev et al. 2000] that HH1(A1 o0)= 0;hence HC1(A1 o0)= 0. Furthermore, it is proved in [Etingof and Ginzburg 2002]for any c that HH1(Ht,c(0)) = 0 for all t ∈ C× except in a countable set. Forall such values of t and c, the Lie algebra sln(Ht,c(0)) has no nontrivial centralextension. For this reason, contrary to [Boyallian et al. 1998], we will not considercentral extensions.

6B. Parabolic subalgebras. In this subsection, we will assume that t 6= 0, so,without loss of generality, let us set t = 1. For a Lie algebra with triangular de-composition, one usually wants to construct representations by induction from itsnonnegative Lie subalgebra (a sort of Borel subalgebra) or, more generally, from abigger subalgebra that contains this one. This suggests that the following definitionmay be relevant.

Definition 6.4. [Boyallian et al. 1998] A parabolic subalgebra q of the Lie algebragln(Ht=1,c(0)) is a graded Lie subalgebra of the form

q=⊕

Zq[k], q[k] = gln(Ht=1,c(0))[k] if k ≥ 0,

q[k] ⊂ gln(Ht=1,c(0))[k] if k < 0.


For k < 0, we can decompose q[k] as

q[k] =⊕

r,l,i, j−rn+ j−i=k

Ei j ur I i,lk el for some subspace I i,l

k ⊂ C[ω].

Lemma 6.5. The subspace I i,lk is an ideal of C[ω].

Proof. Let Ei j ur p(ω)el ∈ q[k] with p(ω) ∈ I i,lk , and choose f (ω) ∈ C[ω]. If

r = 0, then, since Ei i f (ω)el ∈ q[0], we deduce that [Ei i f (ω)el, Ei j p(ω)el] =

Ei j f (ω)p(ω)el ∈ q[k] if 1≤ i 6= j ≤ n; hence f (ω)p(ω) ∈ I i,lk .

Now suppose that r > 0. We want to prove by induction on a ∈ Z≥0 thatωa p(ω) ∈ I i,l

k . We note that

[Iωa+1, Ei j (ur p(ω)el)] = Ei j (ur ((ω− r)a+1−ωa+1)p(ω)el) ∈ q[k]

and that the term of highest power in (ω− r)a+1−ωa+1 is −r(a + 1)ωa , so that

we can apply induction. �

Following the ideas of [Boyallian et al. 1998; Kac and Radul 1993], we choose amonic generator bi,l

k (ω) of the principal ideal I i,lk if this ideal is nonzero; otherwise,

we set bi,lk (ω)= 0. These are called the characteristic polynomials of q.

Definition 6.6. A parabolic subalgebra q is nondegenerate if q[k] 6= 0 for all k ∈Z.

Proposition 6.7. A parabolic subalgebra q is nondegenerate if and only if the poly-nomials bi,l

−1(ω) are all nonzero for 1≤ i ≤ n and 0≤ l ≤ d − 1.

Proof. The parabolic subalgebra q is nondegenerate if and only if the polynomialsbi,l

k (ω) for 1≤ i ≤n and 0≤ l≤d−1 are all nonzero for all k ∈Z≤−1, so it is enoughto prove for i = 1, . . . , n that if bi,l

k (ω) and bi−1,l−1 (ω) are nonzero, then bi,l

k−1(ω)

is also nonzero and divides bi−1,lk (ω)bi,l+r

−1 (ω− r). Here, if 2 ≤ i ≤ n, then r isdetermined by k+ i−1=−rn+ j for some 1≤ j ≤ n. (We set b0,l

k (ω)= bn,lk (ω).)

If i 6= 1, we have[Ei,i−1bi,l+r

−1 (ω)el+r , Ei−1, j ur bi−1,lk (ω)el

]= Ei j ur bi,l+r

−1 (ω− r)bi−1,lk (ω)el − δ j,iδr0 Ei−1,i−1ur bi,l+r

−1 (ω)bi−1,lk (ω)el+r

∈ [q[−1], q[k]]

and [q[−1], q[k]] ⊂ q[k−1], so bi,l+r−1 (ω−r)bi−1,l

k (ω) ∈ I i,lk−1 and the claim is true

if i 6= n.To prove the claim when i = 1 (and with r determined by k =−(r + 1)n+ j),

we consider the commutator[E1nub1,l+r

−1 (ω)el+r , Enj ur bn,lk (ω)el

]= E1 j ur+1b1,l+r

−1 (ω− r)bn,lk (ω)el − δ j1δr,−1 Ennur+1bn,l

k (ω− 1)b1,l+r−1 (ω)el+r ,


which belongs to [q[−1], q[k]] ⊂ q[k− 1], so b1,l+r−1 (ω− r)bn,l

k (ω) ∈ I 1,lk−1.

Using similar computations, one can prove for i = 1, . . . , n that if bi,lk−1(ω) 6= 0,

then bi−1,lk (ω) is nonzero and divides bi,l

k−1(ω). �

The characteristic polynomials bi,l−1(ω) for 1≤ i ≤ n and 0≤ l ≤ d−1 can help

us describe the derived Lie subalgebra [q, q]. Set

gln(Ht=1,c(0))[0,b]

= span{Hiωr bi+1,l−1 (ω)el | 1≤ i ≤ n− 1, r ∈ Z≥0, 0≤ l ≤ d − 1}

⊕ span{E11uv(ω+ 1)r b1,l−1(ω+ 1)el+1− Ennvuωr b1,l

−1(ω)el | r ∈ Z≥0}.

Proposition 6.8. Let b= (bi,l−1(ω))

0≤l≤d−11≤i≤n be the first nd characteristic polynomi-

als of the parabolic subalgebra q. Then

[q, q] =

( ⊕k∈Z,k 6=0

q[k])⊕ gln(Ht=1,c(0))[0,b].

Proof. Since q[k+1]=[q[k], q[1]] if k∈Z≥0, it suffices to show that [q[1], q[−1]]=gln(Ht=1,c(0))[0,b]. We compute

[Ei,i+1ωr el1, Ei+1,i b

i+1,l2−1 (ω)el2] = δl1l2(Ei i − Ei+1,i+1)ω

r bi+1,l2−1 (ω)el1,

[E1nub1,l1−1 (ω)el1, En1v(ω+ 1)r el2] = δl1+1,l2

(E11uv(ω+ 1)r b1,l1

−1 (ω+ 1)el2

− Ennvuωr b1,l1−1 (ω)el1

),

[Ei,i+1ωr el1, E1nub1,l2

−1 (ω)el2] = 0,

[Ei+1,i bi+1,l1−1 (ω)el1, En1vω

r el2] = 0 if 1≤ i ≤ n− 1. �

6C. Embedding into gl∞. One of the main objects used to study the representa-tion theory of gln(A1) in [Boyallian et al. 1998; Kac and Radul 1993] is an em-bedding of the algebra Mn(A1) into the algebra M∞ of infinite matrices with onlyfinitely many nonzero diagonals. This induces an embedding of the Lie algebragln(A1) into gl∞. It comes from the action of gln(A1) on Cn

⊗C[u, u−1]. In this

subsection, we obtain similar embeddings for gln(Ht=1,c(0)) and gln(A1o0)when0 is cyclic. The embedding gln(A1 o0) ↪→ gl∞ is the same as the one consideredin the two papers above when 0 is trivial, and gln(Ht=1,c(0)) ↪→ gl∞ comes alsofrom the action of gln(Ht=1,c(0)) on Cn

⊗C[u, u−1] via the Dunkl embedding of

Ht=1,c(0). (We will reserve the notation M∞ and gl∞ for the algebra and the Liealgebra of infinite matrices with finitely many nonzero entries.)

The space M∞ has a linear basis of elementary matrices Ei j with (i, j)∈Z×Z.The embedding of associative algebras ι : Mn(A1 o0) ↪→ M∞ is given explicitly


by the formula

ι(Ei j usωr ek)=∑l∈Z

(−ld−k)r E(ld+k+s)n+i−1,(ld+k)n+ j−1 for s ∈Z, 0≤k≤d−1.

This restricts to an embedding ι : Mn(Ht=1,c(0)) ↪→ M∞ by pulling back viaHt=1,c(0) ↪→Ht=1,c(0)∼= A1 o0. Explicitly, since

v =−u−1(ω−∑d−1

l=0 cl−1el),

we get

ι(Ei jvek)=∑l∈Z

(ld + k+ ck−1)E(ld+k−1)n+i−1,(ld+k)n+ j−1.

This can be extended to

(7) ι(Ei jvsωr ek)

=

∑l∈Z

(s−1∏p=0

(ld + k− p+ ck−p−1))(−ld − k)r E(ld+k−s)n+i−1,(ld+k)n+ j−1.

The principal grading on the algebra M∞ and on the Lie algebra gl∞ is given bydeg(Ei j )= j − i , and the embedding ι respects all the gradings.

We will need, as in [Boyallian et al. 1998], to consider infinite matrices overthe ring of truncated polynomials Rm = C[t]/(tm+1). Fixing a ∈ C, we define analgebra map ϕ[m]a : Mn(Ht=1,c(0))→ M∞(Rm) by

ϕ[m]a (Ei j uek)=∑l∈Z

E(ld+k+1)n+i−1,(ld+k)n+ j−1,

ϕ[m]a (Ei jvek)=∑l∈Z

(ld + k+ a+ t + ck−1)E(ld+k−1)n+i−1,(ld+k)n+ j−1.

This extends to a map ϕ[m]a : Mn(A1 o0)→ M∞(Rm). Explicitly,

ϕ[m]a (Ei j usωr ek)=∑l∈Z

(−ld − k− a− t)r E(ld+k+s)n+i−1,(ld+k)n+ j−1

for s ∈ Z and 0≤ k ≤ d − 1.This embedding when d = 2 and n = 1 is related to the embedding considered

in [Shoikhet 1998] from the Lie algebra glλ to gl∞,s (in the notation of that paper)since glλ is obtained by turning into a Lie algebra a certain primitive quotient ofUsl2(C) and this primitive quotient is isomorphic to the spherical subalgebra ofHt=1,c=λ(Z/2Z).


6D. Geometric interpretation. Kac and Radul [1993] observed that the algebra ofholomorphic differential operators on C× has a geometric interpretation in termsof a certain infinite-dimensional vector bundle over the cylinder C/Z. The algebrasA1 o0 and A1 o0 afford similar interpretations. To explain it, we have to extendthem to a holomorphic setting.

Let O(w) be the ring of entire functions (holomorphic on all of C) in the vari-able w. Let AO

1 o 0 to be the span of the operators of the form ur f (w)γ withf ∈ O(w) and r ∈ Z. This span has an algebra structure extending the one onA1 o0. Let AO

1 o0 be the subalgebra of AO1 o0 consisting of linear combinations

of operators of the form ur f (w)γ and vs f (w)γ with r, s ≥ 0 and f holomorphic.For k ∈ Z, we define an automorphism θk of M∞ and of gl∞ by θk(Ei j ) =

Ei+k, j+k . For 0 ≤ k ≤ d − 1, let Mk∞⊂ M∞ be the subspace of matrices such

that the (i, j) entry is zero if j 6∈⋃

l∈Z[ldn+ kn, ldn+ (k + 1)n[. The followingdefinition is adapted from [Kac and Radul 1993, Definition 3.4].

Definition 6.9. An (n, d)-monodromic loop is a holomorphic map ` : C→ M∞such that `(w)=`0(w)+· · ·+`d−1(w)with `k(w−d)= θd

n `k(w) and `k(w)∈Mk∞

for 0≤ k ≤ d − 1.

When d=1, the following proposition was established in [Kac and Radul 1993].

Proposition 6.10. The algebra Mn(AO1 o 0) is isomorphic to the algebra Ln,d of

(n, d)-monodromic loops.

Proof. We can construct a map Mn(AO1 o0)→Ln,d by E 7→ (w 7→ ϕ[0]w (E)). That

the formula w 7→ ϕ[0]w (E) defines an (n, d)-monodromic loop follows from theformula for ϕ[0]w . The inverse is given in the following way. If, given a monodromicloop `, the loop `k is concentrated along the (sn+m)-th diagonal (for 0≤m≤n−1)below the main one (so sn+m≥ 0 — if it is above, the argument is similar), so that`k =

∑ni=1

∑l∈Z fi,l,k(w)E(ld+k+s)n+m+i−1,(ld+k)n+i−1, then the preimage of `k is∑n−m

i=1 Em+i,i us fi,0,k(−ω− k)ek +∑m

i=1 Ei,n−m+i us+1 fn−m+i,0,k(−ω− k)ek . �

Since Mn(AO1 o0) ↪→Mn(AO

1 o0), we can identify Mn(AO1 o0)with the algebra

of (n, d)-monodromic loops ` such that, writing `(w) =∑

i, j∈Z `i, j (w)Ei j , wehave that, if i = l1n+ p1− 1, j = l2n+ p2− 1, 1≤ p1, p2 ≤ n and l1 < l2, then`i, j (w)= 0 for w = p− l2d and p = 0, . . . , l2− l1− 1.

7. Highest weight representations for matrix Lie algebras over Cherednikalgebras of rank one

Inspired by the papers [Ginzburg et al. 2003; Guay 2005], we suggest a notion ofcategory O for the Lie algebra sln(Ht=1,c(0)), and we study certain modules in it,which we call quasifinite highest weight modules.


Definition 7.1. Assume that t 6= 0. The category O(sln(Ht=1,c(0))) is the categoryof finitely generated modules M over Usln(Ht=1,c(0)) upon which sln(vC[v]) actslocally nilpotently.

This definition applies also to the 0-deformed double current algebras Dnλ,b(0)

of [Guay 2009b]. One justification for it is that the Schur–Weyl functor studiedin [Guay 2005; 2009b] sends modules in the category O of a rational Cherednikalgebra for 0×l o Sl to a module in O(Dn

λ,b(0)) (for appropriate values of λ,b).It is possible, using induction, to construct analogues of Verma modules in thiscategory, and one can ask about the classification of irreducible (integrable) mod-ules in the category O(sln(Ht=1,c(0))). We will not try to answer this question.Instead, we will study certain modules in these categories by following the ideasin [Boyallian et al. 1998; Kac and Radul 1993].

Recall the grading on sln(A1 o 0) and the embeddings ϕ[m]a : sln(A1 o 0) ↪→

gl∞(Rm). The Lie algebra gl∞ has an obvious triangular structure compatible withthe grading given by deg(Ei j )= j−i , and the embeddings ϕ[m]a respect the gradingon the source and target spaces. The following definition comes naturally from thetriangular structure.

Definition 7.2. Let λi,k,r ∈ C for 1 ≤ i ≤ n, 0 ≤ k ≤ d − 1 and r ∈ Z≥0, andlet λ ∈ gln(H1,c(0))[0]∗ be given by λ(Ei iw

r ek) = λi,k,r . Extending λ to a one-dimensional representation Cλ of the Lie algebra gln(H1,c(0))[≥0] (which is equalto⊕∞

k=0 gln(H1,c(0))[k]) by letting gln(H1,c(0))[k] act trivially if k> 0, we definethe Verma module M(λ) by

M(λ)= Ugln(H1,c(0))⊗Ugln(H1,c(0))[≥0] Cλ.

The following lemma and definition are quite standard.

Lemma 7.3. The Verma module M(λ) has a unique irreducible quotient, whichwe denote by L(λ).

Definition 7.4. We call a gln(H1,c(0))-module a highest weight module of highestweight λ ∈ gln(H1,c(0))[0]∗ if this module is generated by a vector v on whichh∈gln(H1,c(0))[0] acts by multiplication by λ(h) and gln(H1,c(0))[k] acts triviallyif k ∈ Z>0. A vector with this last property is said to be singular.

The highest weight vector which generates the Verma module M(λ) will bedenoted vλ.

Given a parabolic subalgebra q of gln(H1,c(0)) with b the set of its first ndcharacteristic polynomials, one can define similarly generalized Verma modulesM(q, λ) by choosing λ such that λ(h)= 0 for any h ∈ gln(H1,c(0))[0,b], since, inthis case, λ descends to q/[q, q]; see Proposition 6.8.

Our goal now is to study quasifinite irreducible highest weight modules, so weintroduce the next definition.


Definition 7.5. We say a graded highest weight module M =⊕

k∈Z M[k] overgln(H1,c(0)) is quasifinite if dimC M[k]<∞ for all k ∈ Z.

In order to obtain below a condition equivalent to the quasifiniteness of L(λ),we need one more definition, as in [Kac and Radul 1993].

Definition 7.6. The Verma module M(λ) is said to be highly degenerate if thereexists a singular vector v∈M(λ)[−1] such that v= Avλ with A∈gln(H1,c(0))[−1]and qdet(A) 6= 0.

The space gln(H1,c(0))[−1] is spanned by Ei+1,iωr el and by E1nω

r uel , so theentries of a matrix A in gln(H1,c(0))[−1] do not necessarily belong to a commu-tative ring. By qdet(A), we thus mean the quasideterminant of A (which, in thiscase, is, up to a sign, the product of the nonzero entries of A).

Proposition 7.7. The Verma module M(λ) is highly degenerate if and only if λvanishes on gln(H1,c(0))[0,b] for some nd monic (and thus nonzero) polynomialsb= (bi,l(ω))0≤l≤d−1

1≤i≤n .

Proof. The argument from the proof of [Boyallian et al. 1998, Proposition 4.1]applies. The polynomials bi,l(ω) are related to the matrix A as follows. SinceA ∈ gln(Ht=1,c(0))[−1], it can be written as a linear combination of matrices ofthe type Ei+1,i bi+1,l(w)el for 1≤ i ≤ n−1 and E1nb1,l(w)uel with 0≤ l ≤ d−1.It follows from the proof of Proposition 6.8 that gln(H1,c(0))[0,b] is spanned by[B, A] for all B ∈ gln(H1,c(0))[1]. �

Proposition 7.8 [Boyallian et al. 1998]. Given λ ∈ gln(H1,c(0))[0]∗ as before, thefollowing conditions are equivalent:

(1) M(λ) is highly degenerate.

(2) L(λ) is quasifinite.

(3) L(λ) is a quotient of a generalized Verma module M(q, λ) in which all thecharacteristic polynomials b= (bi,l(ω))0≤l≤d−1

1≤i≤n of q are nonzero.

Proof. Proposition 7.7 shows that (1) and (3) are equivalent. Let us show that ifall the polynomials bi,l(ω) are nonzero, then dimC(gln(H1,c(0))[k]/q[k]) is finite;hence L(λ) is quasifinite. Under this assumption, it follows from the proof ofDefinition 6.6 that bi,l

k (w) are nonzero for all k ∈ Z≤−1, 1≤ i ≤ n, 0≤ l ≤ d−1.Recall that, for k < 0, we can write

gln(H1,c(0))[k]=∑

s,l,i, j−sn+ j−i=k

Ei j usC[ω]el and q[k]=∑

s,l,i, j−sn+ j−i=k

Ei j usC[ω]bi,lk (w)el .

Our claim now follows from the observation that dimC(C[w]/(bi,lk (w))) <∞.


Now suppose that L(λ) is quasifinite. Then dimC L(λ)[−1] < ∞, so, withM(λ) the unique maximal submodule of M(λ), we have M(λ)[−1] 6= {0}. All thevectors in M(λ)[−1] 6= {0} are singular and at least one satisfies the condition inDefinition 7.6. Therefore, M(λ) is highly degenerate. �

In Theorems 5.2 and 5.4, we stated a criterion in terms of certain power seriesfor the integrability of the simple quotients of Verma modules for sln(A), sln(B)and sln(C). We now want to give a similar criterion for the quasifiniteness of L(λ).To achieve this, given λ ∈ gln(H1,c(0))[0]∗ as before, set di,l,r = λ(Ei iw

r el) for1 ≤ i ≤ n and Di,l(z) =

∑∞

r=0(di,l,r/r !)zr . Recall that a quasipolynomial is alinear combination of functions of the form p(z)eaz , where p(z) is a polynomialand a ∈ C.

Theorem 7.9. The module L(λ) is quasifinite if and only if there exist quasipoly-nomials φi,l(z) for 1≤ i ≤ n and 0≤ l ≤ d − 1 such that

(1− edz)Di,l(z)={φ1,l(z) if i = 1,φ1,l(z)+ (1− edz)φi,l(z) if 2≤ i ≤ n.

Proof. The proof is similar to the proof of [Boyallian et al. 1998, Theorem 4.1],using the description of gln(H1,c(0))[0,b] given just before Proposition 6.8. Letus explain the differences. Writing bi,l(ω)= ωmi,l + fi,l,mi,l−1ω

mi,l−1+· · ·+ fi,l,0,

we obtain the equations∑mi,l

r=0 fi,l,r Fi,l,r+r = 0 for 1 ≤ i ≤ n and r = 0, 1, . . . ,where Fi,l,r = di,l,r − di−1,l,r for 2 ≤ i ≤ n and fi,l,mi,l = 1. To express F1,l,r interms of the di,l,r , we write

E11uv(ω+ 1)r b1,l−1(ω+ 1)el+1− Ennvuωr b1,l

−1(ω)el

=−E11(ω+ 1)r+1b1,l−1(ω+ 1)el+1+ (cl + 1)E11(ω+ 1)r b1,l

−1(ω+ 1)el+1

+ Ennωr+1b1,l

−1(ω)el − (cl + 1)Ennωr b1,l−1(ω)el

We thus see that

F1,l,r = dn,l,r+1− (cl + 1)dn,l,r −

r+1∑j=0

(r+1j

)d1,l+1, j + (cl + 1)

r∑j=0

( rj

)d1,l+1, j .

Setting Fi,l(z) =∑∞

r=0 Fi,l,r zr/r ! for 1 ≤ i ≤ n, we conclude as in [Boyallianet al. 1998] that Fi,l(z) is a quasipolynomial. For 2≤ i ≤ n, we can write Fi,l(z)=Di,l(z)− Di−1,l(z), and for i = 1, we have

F1,l(z)= D′n,l(z)− (cl + 1)Dn,l(z)− (ez D1,l+1)′(z)+ (cl + 1)ez D1,l+1(z).

Here, D′i,l(z) is the derivative of Di,l(z). This implies that D′i,l(z)− D′i−1,l(z) isalso a quasipolynomial (for 2≤ i ≤ n), and hence so is

(D1,l(z)− ez D1,l+1(z))′− (cl + 1)(D1,l(z)− ez D1,l+1(z)).


Hence ez D1,l+1(z)− D1,l(z) is a quasipolynomial; thus so is (1− edz)D1,l(z). �

It is possible to construct quasifinite representations of gln(H1,c(0)) as tensorproducts of certain modules. This is where the embeddings ϕ[m]a come into play.Unfortunately, they are not necessarily irreducible.

First, we need to construct irreducible representations of gl∞(Rm) using a stan-dard procedure. An element λ ∈ gl∞(Rm)[0]∗ is determined by λ( j)

k = λ(Ekk t j )

for k ∈ Z and j = 0, . . . ,m, which we call its labels, following the terminology in[Boyallian et al. 1998]. Using induction from the subalgebra of upper-triangularmatrices and its one-dimensional representation determined by such a λ, we con-struct a Verma module for gl∞(Rm); this Verma module has a unique irreduciblehighest weight quotient L(m, λ).

The following is [Boyallian et al. 1998, Proposition 4.4].

Proposition 7.10. The irreducible gl∞(Rm)-module L(m, λ) is quasifinite if andonly if for each j = 0, . . . ,m, all but finitely many of the λ( j)

k − λ( j)k+1 are zero.

Let m= (m1, . . . ,m N )∈Z⊕N≥0 and λ= (λ(1), . . . , λ(N )) with λ(i) belonging to

gl∞(Rmi )[0]∗ such that L(mi , λ(i)) is quasifinite. We can form the tensor product

L(m, λ)=⊗N

i=1 L(mi , λ(i)), which is an irreducible quasifinite representation ofgl∞[m] =

⊕Ni=1 gl∞(Rmi ). By pulling it back via the map

ϕ[m]a =

N⊕i=1

ϕ[mi ]ai: gln(Ht=1,c(0))→ gl∞[m] for a= (a1, . . . , aN ) ∈ CN ,

we obtain a representation of gln(Ht=1,c(0)), which we denote by La(m, λ).[Boyallian et al. 1998, Theorem 4.2] does not hold for gln(Ht=1,c(0)), so we

cannot deduce that the representation La(m, λ) is necessarily irreducible. Let usdiscuss what is the difference here. That theorem states that pulling back a quasi-finite representation of gl∞[m] to gln(A1 o0) via ϕ[m]a gives a representation thathas the same submodules. (The proof in the case 0 = {1} extends to any d > 1.)The main ideas of the proof are as follows; see also [Kac and Radul 1993]. First,we have to introduce a holomorphic enlargement of gln(A1 o0), as at the end ofSection 6: It is the Lie algebra gln(AO

1 o0) spanned by Ei j us f (ω)el , where f (ω)is an entire function of ω, the bracket of gln(A1 o 0) extending to gln(AO

1 o 0)

naturally. Second, the formula for the embedding ϕ[m]a (when ai 6= a j for i 6= j)can be used to obtain a map ϕ[m],Oa : gln(AO

1 o 0)→ gl∞[m], which is onto, butnot necessarily into. The last step is to show that, if V is a quasifinite module overgln(A1 o0), then, by continuity, we can make gln(AO

1 o0)[k] act on V if k 6= 0.This involves computing an upper bound on the norm of certain operators.

The first and third step work also for Ht=1,c(0), but the second doesn’t. Considerthe algebra HO

t=1,c(0) spanned by elements of the form vr f (ω)el and us f (ω)el ,


where f (ω) is an entire function of ω and the multiplication is given by (in thecase r ≥ s)

vr f (ω)el1us g(ω)el2= δl1−s,l2vr−s( s∏

k=1

(−ω+cl2+s−k+1+s−k))

f (ω−s)g(ω)el2 .

We have also a map ϕ[m],Oa : gln(HOt=1,c(0)) → gl∞[m], but it is not onto; for

instance, if a = 0 = m = cl for l = 0, . . . , d − 1, then ϕ[m]a (Ei jv f (ω)ek) =∑l∈Z(ld + k) f (−ld − k)E(ld+k−1)n+i−1,(ld+k)n+ j−1. Therefore, in the image, the

coefficient of E−n+i−1, j−1 is always zero, independently of f (ω). At least, wehave the following result.

Proposition 7.11. Assume that ai − a j 6∈ Z for 1≤ i 6= j ≤ N and ck + ai 6∈ Z forall 1≤ i ≤ N and 0≤ k ≤ d − 1. Then ϕ[m],Oa : gln(H

Ot=1,c(0))→ gl∞[m] is onto.

Proof. Decompose gl∞ as gl∞=n−∞⊕h∞⊕n+

∞, where h∞ is the Lie subalgebra of

all the diagonal blocks of size n (with one having a corner at the (0, 0)-entry), andn±∞

are the complements of h∞ consisting of strictly upper and lower triangularmatrices. That ϕ[m],Oa is onto the subspace n−

∞when restricted to the subspace

spanned by the elements Ei j us f (ω)ek with s ∈Z≥0, 1≤ i, j ≤ n and 0≤ k ≤ d−1follows from [Boyallian et al. 1998; Kac and Radul 1993], so let us focus insteadon gln(H

Ot=1,c(0))[> 0]. Explicitly, using the Taylor formula for the expansion of

a function of t around t = 0 and (7), ϕ[mi ]ai is given by

(8) ϕ[mi ]ai(Ei jv

s f (ω)ek)=∑l∈Z

mi∑b=0

g(b)(ai + ld)b!

tb E(ld+k−s)n+i−1,(ld+k)n+ j−1

if we set g(t)=(∏s−1

p=0(k− p+t+ck−p−1))

f (−k−t). As in [Kac and Radul 1993,Proposition 3.1], we can use the fact that, for every discrete set of points in C, thereis a holomorphic function on C with prescribed values of its first mi derivatives ateach points of such a set. Combining this with our assumption that ai − a j 6∈ Z

for 1≤ i 6= j ≤ N , we deduce that, given a matrix E =⊕N

i=1 Ei in gl∞[m], thereexists an entire function g(t) such that the right side of (8) is equal to Ei for alli = 1, . . . , N . To complete the proof, we have to find an entire function f (ω) suchthat, if we set g(t)=

(∏s−1p=0(k− p+ t + ck−p−1)

)f (−k− t), then

g(b)(ai + ld)= g(b)(ai + ld) for 1≤ i ≤ N , 0≤ b ≤ mi and all l ∈ Z.

Set P(t)=∏s−1

p=0(k− p+ t + ck−p−1), so that

g(b)(t)=b∑

a=0

(ba

)P (n−a)(t) f (a)(−k− t).


Fix 1≤ i ≤ N and l ∈ Z and consider the system of equations

g(b)(ai + ld)=b∑

a=0

(ba

)P (b−a)(ai + ld)za for b = 0, 1, . . . ,mi ,

with z0, . . . , zmi being the unknown variables (which we would like to express interms of g(b)(ai+ld)). Our hypothesis that ck+ai 6∈Z implies that P(ai+ld) 6= 0,so the matrix of this system is triangular with nonzero entries along the diagonal.We can thus solve it: Let z0

i,l, z1i,l, . . . , zmi

i,l be a solution. Then we can rephrasethe problem by saying that we now have to find an entire function f (ω) such thatf (a)(ai+ld)= za

i,l for 1≤ i ≤ N , 0≤ a≤mi and all l ∈Z. To deduce the existenceof such a function, we can now apply the same argument as the one used to deducethe existence of g(ω) above. �

The representation La(m, λ) is a highest weight module, so it is interesting tocalculate its associated series Di,k(z), which is equal to

∑Nj=1 Di, j,k(z). The for-

mulas are similar to those in [Boyallian et al. 1998]. Set

h(p)l ( j)= λ(p)l ( j)− λ(p)l+1( j) and g j,l(z)=m j∑p=0

h(p)l ( j)(−z)p/p!.

We have

Di, j,k(z)=m j∑p=1

∑l∈Z

λ(p)(ld+k)n+i−1( j)

(−z)p

p!e−(a j+ld+k)z,

Di, j,k(z)= (1− edz)−1∑l∈Z

e−(a j+ld+k)z(g j,(ld+k)n+i−1(z)+ g j,(ld+k)n+i (z)+ · · ·

+ g j,(ld+k)n+dn+i−2(z))

8. Further discussions

We now present further possible research directions related to the results herein.

8A. Double affine Lie algebras and Kleinian singularities. Let G be an arbitraryfinite subgroup of SL2(C). Such a group G does not always act on the torus C×2

or on C× C×, so we can consider only the algebras C[u, v]o G and C[u, v]G .Moreover, when G is not cyclic, each of these Lie algebras has only one triangulardecomposition, namely

sln(C[u, v]o G)∼= n−(C[u, v]o G)⊕ h(C[u, v]o G)⊕ n+(C[u, v]o G),

sln(C[u, v]G)∼= n−(C[u, v]G)⊕ h(C[u, v]o G)⊕ n+(C[u, v]G)

These also admit universal central extensions. Since C[u, v]G is commutative,HC1(C[u, v]G) = �1(C[u, v]G)/d(C[u, v]G). We know from [Kassel 1984] that


the bracket on the universal central extension

sln(C[u, v]G)= sln(C[u, v]G)⊕ (�1(C[u, v]G)/d(C[u, v]G))

of sln(C[u, v]G) is given by

[m1⊗ p1,m2⊗ p2] = [m1,m2]⊗ (p1 p2)+Tr(m1m2)p1dp2.

As for sln(C[u, v] o G), it is known that its kernel HC1(C[u, v] o G) is equalto �1(C[u, v])G/d(C[u, v]G); see [Farinati 2005]. However, to obtain an explicitformula for its bracket, one would have to choose a splitting

〈C[u, v]o G,C[u, v]o G〉 = [C[u, v]o G,C[u, v]o G]⊕HC1(C[u, v]o G);

see Section 2.In [Kapranov and Vasserot 2000], the authors proved that the derived category

of coherent sheaves on the minimal resolution C2/G of the singularity C2/G isequivalent to the derived category of modules over the skew-group ring C[u, v]oG.It is thus natural to ask if there is a connection between the derived category ofrepresentations of sln(C[u, v] o G) and the derived category of modules over acertain sheaf of Lie algebras on C2/G.

In the same line of thought, since A1 o G and AG1 are Morita equivalent, one

can wonder about the connections between sln(A1 o G) and sln(AG1 ). However,

even if A and B are Morita equivalent rings, the categories of representations ofsln(A) and sln(B) are not necessarily equivalent. As a counterexample, one canconsider A = C[t±1

] and B = C[u±1] o (Z/dZ) ∼= Md(C[t±1

]), in which casesln(B)= slnd(C[t±1

]).The definitions of Weyl modules recalled in Section 5B can be adapted to both

sln(C[u, v]oG) and sln(C[u, v]G). Studying these appears to be a reasonable wayto approach the representation theory of these Lie algebras since, when G is notcyclic, we do not have triangular decompositions similar to (3) or presentationsas in Proposition 5.3. It would be interesting to compute the dimension of localWeyl modules at the Kleinian singularity. For smooth points on an affine varietyand certain highest weights, the dimension of local Weyl modules was computedin [Feigin and Loktev 2004], and [Kuwabara 2006] treated the case of a doublepoint. One can expect the study of such local Weyl modules to be related to thegeometry of the minimal resolution of the Kleinian singularity.

8B. Quiver Lie algebras. Symplectic reflection algebras for wreath products ofG are known to be Morita equivalent to certain deformed preprojective algebrasof affine Dynkin quivers, which are called Gan–Ginzburg algebras in the literature[Gan and Ginzburg 2005]. In the rank one case, these are the usual deformedpreprojective algebras5λ(Q). The affine Dynkin diagram in question is associated


to G via the McKay correspondence. The quantum Lie algebra analogues of theseGan–Ginzburg algebras were introduced in [Guay 2009a] and are deformations ofthe enveloping algebra of a Lie algebra which is slightly larger than the universalcentral extension of sln(5(Q)), where 5(Q)=5λ=0(Q). The same themes as inthe previous sections can be studied in the context of the Lie algebra sln(5(Q)),in particular when the graph underlying Q is an affine Dynkin diagram. Actually,when Q is the cyclic quiver on d vertices, 5(Q)∼=C[u, v]o0. Furthermore, if e0

is the extending vertex of an affine Dynkin diagram, then e05(Q)e0 ∼= C[u, v]G .All these are examples of matrix Lie algebras over interesting noncommutative

rings. It is possible to replace sln by another semisimple Lie algebra. This isexplained in [Berenstein and Retakh 2008]. It would also be interesting to compareour work with the constructions in [Halbout et al. 2008].

Acknowledgments

Guay gratefully acknowledges the hospitality of the Laboratoire de Mathematiquesde l’Universite de Versailles-St-Quentin-en-Yvelines where this project was startedwhile he was a postdoctoral researcher supported by the Ministere francais de laRecherche. He is also grateful for the support received from the University ofEdinburgh, the University of Alberta and an NSERC Discovery Grant. Loktevwas partially supported by RF President Grant N.Sh-3035.2008.2, grants RFBR-08-02-00287, RFBR-CNRS-07-01-92214 and RFBR-IND-0801-91300, and the P.Deligne 2004 Balzan prize in mathematics. We thank A. Pianzola for pointing outthe reference [Berman et al. 2003]. We are grateful to I. Gordon for his comments,for his suggestions regarding Corollary 5.21 and for pointing out the counterexam-ple from the work of M. Haiman that we presented at the end of Section 5C.

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Received January 6, 2009. Revised June 13, 2009.

NICOLAS GUAY

DEPARTMENT OF MATHEMATICAL AND STATISTICAL SCIENCES

UNIVERSITY OF ALBERTA

CAB 632EDMONTON, ALBERTA T6G 2G1CANADA

[email protected]

DAVID HERNANDEZ

CNRS - ECOLE NORMALE SUPERIEURE

45, RUE D’ULM

75005 PARIS

FRANCE

[email protected]

SERGEY LOKTEV

INSTITUTE FOR THEORETICAL AND EXPERIMENTAL PHYSICS

MOSCOW 117218RUSSIA

[email protected]

http://www.ams.org/mathscinet-getitem?mr=2004e:17012




http://dx.doi.org/10.1007/s00222-005-0495-3











http://dx.doi.org/10.1007/s002220050242

http://dx.doi.org/10.1007/s002220050242



http://dx.doi.org/10.1016/S0021-8693(02)00073-X







COMPARISON RESULTS FOR CONJUGATE AND FOCALPOINTS IN SEMI-RIEMANNIAN GEOMETRY

VIA MASLOV INDEX

MIGUEL ÁNGEL JAVALOYES AND PAOLO PICCIONE

We prove an estimate on the difference of Maslov indices relative to thechoice of two distinct reference Lagrangians of a continuous path in theLagrangian Grassmannian of a symplectic space. We discuss some applica-tions to the study of conjugate and focal points along a geodesic in a semi-Riemannian manifold.

1. Introduction

Classical comparison theorems for conjugate and focal points in Riemannian orcausal Lorentzian geometry require curvature assumptions, or Morse index argu-ments; see [Ambrose 1957; Eschenburg and O’Sullivan 1980; Galloway 1979;Kupeli 1986; 1988]. In the general semi-Riemannian world, this approach doesnot work, because the curvature is never bounded [Dajczer and Nomizu 1980]and the index form has always infinite Morse index. In addition, it is well knownthat singularities of the semi-Riemannian exponential map may accumulate along ageodesic [Piccione and Tausk 2003], and there is no hope to formulate a meaningfulcomparison theorem using assumptions on the number of conjugate or focal points.

There are several good indications that a suitable substitute for the notion of sizeof the set of conjugate or focal points along a semi-Riemannian geodesic is givenby the Maslov index. This is a symplectic integer-valued invariant associated tothe Jacobi equation, or more generally to the linearized Hamilton equations alongthe solution of a Hamiltonian system. This number replaces the Morse index of theindex form, which in the general semi-Riemannian case is always infinite, and insome nondegenerate cases it is a sort of algebraic count of the conjugate points. In

MSC2000: 53C22, 53C50, 53D12, 53D25.Keywords: Maslov index, geodesics, semi-Riemannian geometry, conjugate points, focal points.This work was partially supported by MEC project MTM2007-64504, and Fundación Séneca project04540/GERM/06, Spain. This research was undertaken within the Programme in Support of Excel-lence Groups of the Región de Murcia, Spain, by Fundación Séneca, Regional Agency for Scienceand Technology (Regional Plan for Science and Technology 2007-2010). Javaloyes is partially spon-sored by Regional J. Andalucía Grant P06-FQM-01951 and Piccione by Capes (Brazil), Grant BEX1509-08-0, and Fundación Séneca grant 09708/IV2/08, Spain.

43



44 MIGUEL ÁNGEL JAVALOYES AND PAOLO PICCIONE

the Riemannian or causal Lorentzian case, the Maslov index of a geodesic relativeto some fixed Lagrangian coincides with the number of conjugate (or focal) pointscounted with multiplicity. The exponential map is not locally injective around non-degenerate conjugate points [Warner 1965], or more generally around conjugatepoints whose contribution to the Maslov index is nonzero [Piccione et al. 2004].

Inspired by [Lytchak 2008], in this paper we prove an estimate on the differencebetween Maslov indices (Proposition 3.3), and we apply this estimate to obtain anumber of results that are the semi-Riemannian analogue of the standard compar-ison theorems in Riemannian geometry (Section 4). These results relate the exis-tence and the multiplicity of conjugate and focal points with the values of Maslovindices naturally associated to a given geodesic. It is very interesting to observe thatRiemannian versions of the results proved in the present paper, which are mostlywell known, are obtained here with a proof that appears to be significantly moreelementary than the classical proof using Morse theory. Detecting conjugate pointsis interesting because their presence implies multiplicity of geodesics between pairsof points. Estimates on the number of geodesics in terms of conjugate points canbe done either via Morse theory or via bifurcation theory. Recent results in thisdirection are available also in the case of non-Riemannian metrics; see for instance[Abbondandolo et al. 2003; Abbondandolo and Majer 2008] for the Morse theoryof geodesics in globally hyperbolic Lorentzian manifolds, [Giannoni et al. 2001]for that theory in stationary Lorentzian manifolds, and [Piccione et al. 2004] forthe bifurcation theory of geodesics in arbitrary semi-Riemannian manifolds.

As to the applications in Morse theory or bifurcation theory of the results of thepresent paper, an important observation is in order. It is now well established thatin semi-Riemannian geometry there exist conjugate points that do not contributeto the spectral flow of the second variation of the geodesic action functional. Thespectral flow is an integer-valued invariant associated to continuous paths of sym-metric Fredholm bilinear forms, and it is a natural substitute for the Morse index instrongly indefinite variational problems. Is is known that jumps of the Maslov in-dex detect bifurcation of geodesics [Piccione et al. 2004]. Similarly, for the Morsetheory of geodesics, only conjugate points that contribute to the spectral flow arerelevant. The conjugate points whose existence is established in the results of thispaper are detected via jumps of the Maslov index, which by the semi-RiemannianMorse index theorem is equal to the spectral flow. Thus, our results can indeed beused in Morse theory and in bifurcation theory.

The paper is organized as follows. In Section 2 we recall a few basic facts aboutthe geometry of the Lagrangian Grassmannian3 of a symplectic space (V, ω), andabout the notion of Maslov index for continuous paths in 3. We use a generalizednotion of Maslov index, which applies to paths with arbitrary endpoints; for pathswith endpoints on the Maslov cycle, there are several conventions regarding the

COMPARISON RESULTS FOR CONJUGATE AND FOCAL POINTS 45

contribution of the endpoints. Here we adopt a convention slightly different fromthat in [Robbin and Salamon 1993]; see (2-3), (2-4) and (2-5).

Section 3 contains the estimate (3-1) on the difference of Maslov indices relativeto the choice of two arbitrarily fixed reference Lagrangians L0 and L1. Usingthe canonical atlas of charts of the Grassmannian Lagrangian and the transitionmap (2-1), the proof is reduced to studying the index of perturbations of symmetricbilinear forms; see Lemma 3.1 and Corollary 3.2. Several analogous estimates,(3-2) and (3-3), are obtained using properties (2-4) and (2-6) of Hormander’s index.

Section 4 discusses applications to the study of conjugate and focal points alongsemi-Riemannian geodesics. In Section 4.1, we describe how to obtain Lagrangianpaths out of the flow of the Jacobi equation along a geodesic γ : [a, b] → M andan initial nondegenerate submanifold P of a semi-Riemannian manifold (M, g).In Lemma 4.1, we give a characterization of which Lagrangian subspaces of thesymplectic space Tγ(a)M⊕ Tγ(a)M arise from an initial submanifold construction.Section 4.1 proves the comparison results, which include comparison between con-jugate and focal points, as well as between conjugate points relative to distinctinitial endpoints. We conclude the paper in Section 5 with a few final remarksconcerning the question of nondegeneracy of conjugate and focal points.

2. Preliminaries

The Lagrangian Grassmannian. Let us consider a symplectic space (V, ω), withdim(V ) = 2n; we will denote by Sp(V, ω) the symplectic group of (V, ω), whichis the closed Lie subgroup of GL(V ) consisting of all isomorphisms that preserveω. A subspace X ⊂ V is isotropic if the restriction of ω to X × X vanishes iden-tically; an n-dimensional (that is, maximal) isotropic subspace L of V is called aLagrangian subspace. We denote by 3 the Lagrangian Grassmannian of (V, ω),which is the collection of all Lagrangian subspaces of (V, ω), and is a compact dif-ferentiable manifold of dimension 1

2 n(n+1). A real-analytic atlas of charts on3 isgiven as follows. Start with a Lagrangian decomposition (L0, L1) of V , that is, onein which L0, L1 ∈3 are transverse Lagrangians with V = L0⊕L1. Then denote by30(L1) the open and dense subset of 3 consisting of all Lagrangians L transverseto L1. A diffeomorphism ϕL0,L1 from 30(L1) to the vector space Bsym(L0) of allsymmetric bilinear forms on L0 is defined by ϕL0,L1(L)= ω(T · , · )|L0×L0 , whereT : L0 → L1 is the unique linear map whose graph in L0 ⊕ L1 = V is L . Thekernel of ϕL0,L1(L) is the space L ∩ L0.

We will need to express the transition map ϕL1,L◦ϕ−1L0,L , where L0, L1, L ∈3 are

three Lagrangians such that L ∩ L0 = L ∩ L1 = {0}. Note that the two charts ϕL0,L

and ϕL1,L have the same domain. If η : L1→ L0 denotes the isomorphism definedas the restriction to L1 of the projection L ⊕ L0→ L0, then for all B ∈ Bsym(L0)


we have (see for instance [Piccione and Tausk 2008, Lemma 2.5.4])

(2-1) ϕL1,L ◦ϕ−1L0,L(B)= η

∗B+ϕL1,L(L0),

where η∗ is the pull-back by η.If (L0, L1) is a Lagrangian decomposition of V , there is a bijection between 3

and the set of pairs (P, S), where P ⊂ L1 is a subspace and S : P × P → R isa symmetric bilinear form on P; see [Piccione and Tausk 2008, Exercise 1.17].More precisely, to each pair (P, S) one associates the Lagrangian subspace L P,S

defined by

(2-2) L P,S ={v+w : v ∈ P, w ∈ L0, ω(w, · )|P + S(v, · )= 0

}.

Maslov index. Let us recall a few notions related to symmetric bilinear forms;for further details we recommend [Piccione and Tausk 2008]. Given a symmetricbilinear form B on a (finite-dimensional) real vector space W , the index of B isdefined to be the dimension of a maximal subspace of W on which B is negativedefinite. The coindex of B is the index of −B, and the signature sign(B) of B isdefined to be the difference coindex minus index.

We will now recall briefly the notion of Maslov index for a continuous path` : [a, b] → 3. For a fixed Lagrangian L0 ∈ 3, the L0-Maslov index µL0(`) of `is the unique integer such that

(a) µL0 is fixed-endpoint homotopy invariant;

(b) µL0 is additive by concatenation;

(c) if `([a, b])⊂30(L1) for some Lagrangian L1 transverse to L0, then

(2-3) µL0(`)= n+[ϕL0,L1(`(b))] − n+[ϕL0,L1(`(a))].

See [Giambo et al. 2004] for a similar discussion. Let us denote by µ−L0the L0-

Maslov index function relative to the opposite symplectic form −ω on V . Therelation between the functions µL0 and µ−L0

is given by

(2-4) µ−L0(`)=−µL0(`)+ dim(`(a)∩ L0)− dim(`(b)∩ L0),

for every continuous path ` : [a, b] →3.Let us emphasize that, for curves ` whose endpoints are not transverse to L0,

there are several conventions for how the endpoints contribute to the Maslov index.For instance, the definition of L0-Maslov index µL0 in [Robbin and Salamon 1993]is obtained by replacing (2-3) with

(2-5) µL0(`)=12 sign[ϕL0,L1(`(b))] −

12 sign[ϕL0,L1(`(a))],

in which case the Maslov index takes values in 12 Z.1

1In this convention, the Maslov index changes sign when one takes the opposite symplectic form.


Given any continuous path ` : [a, b]→3 and any two Lagrangians L0, L ′0 ∈3,the difference µL0(`) − µL ′0(`) depends only on L0 and L ′0 and the endpoints`(a) and `(b) of `. This quantity will be denoted by q(L0, L ′0; `(a), `(b)), and itcoincides (up to some factor that is irrelevant here) with the so-called Hormanderindex; see [Hormander 1971, Definition 3.3.2]. The Hormander index satisfiescertain symmetries; we will need that

(2-6) q(L0, L1; L ′0, L ′1)=−q(L ′0, L ′1; L0, L1) for all L0, L1, L ′0, L ′1 ∈3.

The quantity τ(L0, L1, L2) = q(L0, L1; L2, L0) = −q(L0, L1; L0, L2) coincides(up to some factor) with the Kashiwara index [Lion and Vergne 1980]. The Kashi-wara index function determines completely the Hormander index by the identity

(2-7) q(L0, L1; L ′0, L ′1)= τ(L0, L1, L ′0)− τ(L0, L1, L ′1)

for all L0, L1, L ′0, L ′1 ∈3,

which is easily proved using property (b) of the Maslov index.

3. An estimate of the difference of Maslov indices

Lemma 3.1. Let B and C be symmetric bilinear forms on a (finite dimensional)real vector space V . Then −n−(C)≤ n+(B+C)− n+(B)≤ n+(C).

Proof. It suffices to prove the inequality n+(B+C)−n+(B)≤ n+(C); if this holdsfor every B and C , replacing C with −C and B with B +C will yield the otherinequality −n−(C) ≤ n+(B +C)− n+(B). Choose W ⊂ V a maximal subspaceof V on which B+C is positive definite, so that dim(W )= n+(B+C), and writeW = W+ ⊕ W−, where B|W+×W+ is positive definite and B|W−×W− is negativesemidefinite. Since B+C is positive definite on W , it follows that C |W−×W− mustbe positive definite, so that n+(C |W×W )≥ dim(W−). Then

n+(B+C)= dim(W )= dim(W−)+ dim(W+)

≤ n+(C |W×W )+ n+(B|W×W )≤ n+(B)+ n+(C). �

Corollary 3.2. Let C be a fixed symmetric bilinear form on V . Then for allB1, B2 ∈ Bsym(V ),∣∣n+(B1)− n+(B2)− n+(B1+C)+ n+(B2+C)

∣∣≤ n−(C)+ n+(C).

Proposition 3.3. For any continuous curve ` : [a, b]→3 and any pair L0, L1 ∈3

of Lagrangians, we have

(3-1)∣∣µL0(`)−µL1(`)

∣∣≤ n− dim(L0 ∩ L1).


Proof. Since µL0(`)−µL1(`) depends only on the endpoints `(a) and `(b), we canassume the existence of a Lagrangian L ∈30(L0)∩3

0(L1) such that `(t)∈30(L)for all t ∈ [a, b]. Namely, we choose L ∈30(L0)∩3

0(L1)∩30(`(a))∩30(`(b))

(these are dense opens subsets of 3, hence their intersection is nonempty!), andreplace ` by any continuous curve in 30(L) from `(a) to `(b).

Once we are in this situation, then the Maslov indices of ` are given by

µL0(`)= n+[ϕL0,L(`(b))] − n+[ϕL0,L(`(a))],

µL1(`)= n+[ϕL1,L(`(b))] − n+[ϕL1,L(`(a))].

Now consider the isomorphism η : L1 → L0 obtained as the restriction to L1 ofthe projection L⊕ L0→ L0; using formula (2-1) for the transition function for thecharts ϕL0,L and ϕL1,L , we have for all α ∈30(L)

ϕL1,L(α)= η∗(ϕL0,L(α)+ η∗ϕL1,L(L0)),

and so n+(ϕL1,L(α))=n+(ϕL0,L(α)+C), where C=η∗ϕL1,L(L0) does not dependon α. Note that

n+(C)+ n−(C)= n− dim(Ker(C))= n− dim(L0 ∩ L1).

Inequality (3-1) is obtained easily from Corollary 3.2 by setting B1 = ϕL0,L(`(b))and B2 = ϕL0,L(`(a)). �

Using the symmetry property (2-6) of Hormander index, we also get this estimate:

Corollary 3.4. For any continuous curve ` : [a, b] →3 and any pair L0, L1 ∈3

of Lagrangians, we have

(3-2)∣∣µL0(`)−µL1(`)

∣∣≤ n− dim(`(a)∩ `(b)).

Moreover, changing the sign of the symplectic form and using (2-4), one obtainseasily the inequalities

(3-3)

∣∣µL0(`)−µL1(`)− dim(`(a)∩ L0)+ dim(`(a)∩ L1)

+ dim(`(b)∩ L0)− dim(`(b)∩ L1)∣∣≤ n− dim(L0 ∩ L1),∣∣µL0(`)−µL1(`)− dim(`(a)∩ L0)+ dim(`(a)∩ L1)

+ dim(`(b)∩ L0)− dim(`(b)∩ L1)∣∣≤ n− dim(`(a)∩ `(b)).

4. Comparison results for conjugate and focal points

4.1. Geodesics and Lagrangian paths. Let us now look specifically at curves ofLagrangians arising from the Jacobi equation along a semi-Riemannian geodesic.Let (M, g) be a semi-Riemannian manifold of dimension n, and let ∇ be the co-variant derivative of the Levi-Civita connection of g, with curvature tensor chosen


with the sign convention R(X, Y )=[∇X ,∇Y ]−∇[X,Y ]. We will assume throughoutthe section that γ : [a, b]→M is a given geodesic in M ; when needed, we will alsoconsider extensions of γ to a larger interval [a′, b′] ⊃ [a, b]. The Jacobi equationalong γ is given by (D/dt)2V − R(γ, V )γ = 0. Consider the flow of the Jacobiequation, which is the family of isomorphisms

8t : Tγ(a)M ⊕ Tγ(a)M→ Tγ(t)M ⊕ Tγ(t)M for t ∈ [a, b]

defined by 8t(v,w) = (Jv,w(t), (D/dt)Jv,w(t)), where Jv,w is the unique Jacobifield along γ satisfying J (a) = v and (D/dt)J (a) = w. Consider the symplec-tic form ω on the space V = Tγ(a)M ⊕ Tγ(a)M given by ω((v1, w1), (v2, w2)) =

g(v2, w1)−g(v1, w2). For all t ∈[a, b], define L t0={0}⊕Tγ(t)M⊂Tγ(t)M⊕Tγ(t)M

and set `(t)=8−1t (L t

0). An immediate calculation shows that `(t) is a Lagrangiansubspace of (V, ω), and we obtain in this way a smooth curve ` : [a, b]→3(V, ω).Note that

(4-1) `(a)= La0 =: L0.

Now, consider a smooth connected submanifold P ⊂ M , with γ(a) ∈ P andγ(a) ∈ Tγ(a)P⊥.2 Let us also assume that P is nondegenerate at γ(a), meaningthat the restriction of the metric g to Tγ(a)P is nondegenerate.3 We will denote byn−(g,P) and n+(g,P) respectively the index and the coindex of the restriction ofg to P, so that n−(g,P)+ n+(g,P) = dim(P). Let S be the second fundamentalform of P at γ(a) in the normal direction γ(a), seen as a g-symmetric operatorS : Tγ(a)P → Tγ(a)P. We say that a Jacobi field is P-Jacobi if V (a) ∈ Tγ(a)Pand V ′(a)+ S[V (a)] ∈ Tγ(a)P⊥. An instant t0 ∈ (a, b] is P-focal if there exists anonzero P-Jacobi field vanishing at t0. The multiplicity of a P-focal instant t0 is thedimension of the space of P-Jacobi fields vanishing at t0. Consider the subspaceLP ⊂ V defined by

LP ={(v,w) ∈ Tγ(a)M ⊕ Tγ(a)M : v ∈ Tγ(a)P, w+ S(v) ∈ Tγ(a)P⊥

},

which is just the construction of Lagrangian subspaces described abstractly in (2-2).If π1 : Tγ(a)M ⊕ Tγ(a)M→ Tγ(a)M is the projection onto the first summand, thenπ1(LP)= Tγ(a)P is orthogonal to γ(a). Conversely:

Lemma 4.1. Let L ⊂ Tγ(a)M ⊕ Tγ(a)M be a Lagrangian subspace, and assumethat P = π1(L) is orthogonal to γ(a). Then, there exists a smooth submanifold P

orthogonal to γ(a) such that L = LP.

2The symbol ⊥ denotes orthogonality with respect to the semi-Riemannian metric g.3The assumption of nondegeneracy for the initial submanifold is not strictly necessary for most

of the results of the paper. This assumption is used here for two reasons. First, it guarantees thatthere are no P-focal points on an initial portion of the geodesic γ. Second, it allows us to write thesecond fundamental form as a symmetric linear operator on Tγ(a)P


Proof. Consider the Lagrangian decomposition (L0, L1) of Tγ(a)M⊕Tγ(a)M givenby L0 = {0} ⊕ Tγ(a)M and L1 = Tγ(a)M ⊕ {0}. Then there exists a symmetricbilinear form S : P × P→ R such that L = L P,S as in (2-2). Let P0 ⊂ Tγ(a)M bethe submanifold given by the graph of the function P 3 x 7→ 1

2 S(x, x)γ(a) ∈ P⊥.The desired submanifold P is obtained by taking the exponential of a small openneighborhood of 0 in P0. It is easily seen that the tangent space to P0 at 0 is P ,and since d expγ(a)(0) is the identity, Tγ(a)P= P . Moreover, using the fact that theChristoffel symbols of the chart expγ(a) vanish at 0, it is easily seen that the secondfundamental form of P at γ(a) in the normal direction γ(a) is S. �

Let us also consider the space L0 = {0} ⊕ Tγ(a)M , which corresponds to theLagrangian associated to the trivial initial submanifold P={γ(a)}. Then, an instantt ∈ ]a, b] is P-focal along γ if and only if `(t)∩ LP 6= {0}, and the dimension ofthis intersection equals the multiplicity of t as a P-focal instant. In particular, t isa conjugate instant, that is, γ(t) is conjugate to γ(a) along γ if `(t) ∩ L0 6= {0}.Note that

(4-2) L0 ∩ LP = {0}⊕ Tγ(a)P⊥.

Thus

(4-3) dim(L0 ∩ LP)= codim(P).

For all t ∈ ]a, b], consider the space

AP[t] ={(D/dt)J (t) : J is a P-Jacobi field along γ with J (t)= 0

},

while for t = a we set AP[a] = Tγ(a)P⊥. Note that dim(AP[t])= dim(`(t)∩ LP).When the initial submanifold is just a point, we will use the notation

(4-4)

A0[t] ={(D/dt)J (t) : J is a Jacobi field along γ

with J (a)= 0 and J (t)= 0},

A0[a] = Tγ(a)M.

It is well known that focal or conjugate points along a semi-Riemannian geodesicmay accumulate [Piccione and Tausk 2003]; however, nondegenerate conjugate orfocal points are isolated. A P-focal point γ(t) along γ is nondegenerate when therestriction of the metric g to the space AP[t] is nondegenerate. This is always thecase when g is positive definite (that is, Riemannian), or if g has index 1 (that is,Lorentzian) and γ is either timelike or lightlike. Also, the initial endpoint γ(a),which is always P-focal of multiplicity equal to the codimension of P, is alwaysisolated.

For all t ∈ [a, b], let us denote by n−(g,P, t), n+(g,P, t) and σ(g,P, t) re-spectively the index, the coindex and the signature of the restriction of g to AP[t].


If γ(t) is a nondegenerate P-focal point along γ with t ∈ ]a, b[, then t is an isolatedinstant of nontransversality of the Lagrangians `(t) and LP. Its contribution to theMaslov indexµLP(`), that is, µLP(`|[t−ε,t+ε])with ε>0 sufficiently small, is givenby the integer σ(g,P, t). The contribution of the initial point to the Maslov indexµLP(`), which as observed is always nondegenerate, is given by n+(g,P, a):

(4-5) µLP(`|[a,a+ε])= n+(g,P, a)= n+(g)− n+(g,P).

In particular,

(4-6) µL0(`|[a,a+ε])= n+(g).

Moreover, if γ(b) is a nondegenerate P-focal point along γ, then its contributionto µLP(`) is equal to −n−(g,P, b). Thus, when g is Riemannian the Maslovindex µLP(`|[a+ε,b]) is the number of P-focal points along γ|[a,b[ counted withmultiplicity. The same holds when g is Lorentzian (that is, index equal to 1) and γis timelike. More generally, if all P-focal points along γ are nondegenerate, theMaslov index µLP(`) is given by the finite sum

µP(`)= n+(g)− n+(g,P)+∑

t∈]a,b[

σ(g,P, t)− n−(g,P, b).

All this follows easily from the following elementary result:

Lemma 4.2. Let B : I → Bsym(V ) be a C1-curve of symmetric bilinear forms ona real vector space V . Assume that t0 ∈ I is a degeneracy instant, and denote byB0 the restriction to Ker(B(t0)) of the derivative B ′(t0). If B0 is nondegenerate,then t0 is an isolated degeneracy instant, and for ε > 0 sufficiently small,

n+(B(t0+ ε))− n+(B(t0))= n+(B0), n+(B(t0))− n+(B(t0− ε))=−n−(B0).

Lemma 4.2 is used to compute the Maslov index µLP as follows. Given aP-focal instant t0 ∈ [a, b] and a Lagrangian L1 transversal to both LP and `(t0),consider the smooth path t 7→ ϕLP,L1(`(t)) of symmetric bilinear forms on LP.The kernel of B(t0) is identified with the space AP[t0], and the restriction of thederivative B ′(t0) to Ker(B(t0)) with the restriction of the metric g to AP[t0]; seefor instance [Mercuri et al. 2002].

Comparison results. Let us now prove some comparison results for conjugate andfocal instants.

Proposition 4.3. Given any interval [α, β] ⊂ [a, b],

(4-7)∣∣µL0(`|[α,β])−µLP(`|[α,β])

∣∣≤ dim(P).

Proof. It follows readily from Proposition 3.3 and (4-3). �


In particular, we have the following result concerning the existence of conjugateor focal instant along an arbitrary portion of a geodesic.

Corollary 4.4. For any interval [α, β] ⊂ ]a, b],

• if |µL0(`|[α,β])|> dim(P), then there is at least one P-focal instant in [α, β];

• if |µLP(`|[α,β])| > dim(P), then there is at least one conjugate instant in[α, β].

Proof. By Proposition 4.3, if |µL0(`|[α,β])| > dim(P), then |µLP(`|[α,β])| > 0.Since a 6∈ [α, β], this implies that there is a P-focal instant in [α, β]. The secondstatement is totally analogous. �

On the other hand, the absence of conjugate (focal) instants gives an upper boundon the number of focal (conjugate) instants.

Proposition 4.5. If γ has no conjugate instant, then∣∣µLP(`|[α,β])∣∣≤ dim(P) for every interval [α, β] ⊂ ]a, b].

Similarly, if γ has no P-focal instant, then |µL0(`|[α,β])| ≤ dim(P).

Proof. If γ has no conjugate (respectively, P-focal) instant, then the Maslov indexµL0(`|[α,β])= 0 (respectively, µLP(`|[α,β])= 0) for all [α, β] ⊂ ]a, b]. �

All the statements above have a much more appealing version in the Riemannianor timelike Lorentzian case, where the “Maslov index” can be replaced by thenumber of conjugate or focal instants. In this situation, focal and conjugate instantsare always nondegenerate and isolated, and without using Morse theory one canprove nice comparison results:

Corollary 4.6. Assume that either g is Riemannian or that g is Lorentzian and γis timelike (in which case P is necessarily a spacelike submanifold of M). Denoteby t0 and tP the instants

t0 = sup{t ∈ ]a, b] : there are no conjugate instants in ]a, t]

},

tP = sup{t ∈ ]a, b] : there are no P-focal instants in ]a, t]

}.

Then tP ≤ t0, and if tP = t0, then the multiplicity of tP as a P-focal point is greaterthan or equal to its multiplicity as a conjugate point.

Proof. Assume t0 < tP ≤ b and choose t ′ ∈ ] t0, tP[. Since there are no P-focal in-stants in ]a, t ′] and P is spacelike, (4-5) implies µLP(`|[a,t ′])= codim(P)−n−(g).On the other hand, µL0(`|[a,t ′])≥ n+(g)+ 1 since t0 is conjugate. Hence,

µL0(`|[a,t ′])−µLP(`|[a,t ′])≥ n+(g)+ n−(g)− codim(P)+ 1= dim(P)+ 1,

which contradicts (4-7).


Assume that tP = t0 and that tP is a P-focal point. By possibly extendingthe geodesic γ to a slightly larger interval [a, b′] with b′ > b, we can assume theexistence of t ′> tP with the property that there are no conjugate or P-focal instantsin ]tP, t ′]. Then µL0(`|[a,t ′]) = n+(g)+mul(tP), where mul(tP) is the (possiblynull) multiplicity of tP as a conjugate instant. Similarly

µLP(`|[a,t ′])= codim(P)− n−(g)+mulP(tP),

where mulP(tP) is the multiplicity of tP as a P-focal instant. Then

µL0(`|[a,t ′])−µLP(`|[a,t ′])= dim(P)+mul(tP)−mulP(tP)

which has to be less than or equal to dim(P), giving mul(tP)≥mulP(tP). �

It is known that the result of Corollary 4.6 does not hold without the assumptionthat the metric g is positive definite or that g is Lorentzian and γ timelike. Kupeli[1988, remark in page 585] gives a counterexample by constructing a spacelikegeodesic γ orthogonal to a timelike submanifold P of a Lorentzian manifold, withthe property that γ has conjugate points but no focal point.

In the following statements, ε will denote a small positive number with theproperty that there are no conjugate or P-focal instants in ]a, a+ ε].

Proposition 4.7. We have −n−(g,P)≤ µLP(`|[a+ε,b])−µL0(`|[a+ε,b])≤ dim P.

Proof. This is a straightforward consequence of formulas (4-5), (4-6) and (4-7)applied on the intervals [a, b] and [a+ ε, b]. �

In particular, when g is Riemannian, or g is Lorentzian and γ timelike, thisproposition says that the number of P-focal points along γ is greater than or equalto the number of conjugate points along γ, and that their difference is less than orequal to the dimension of P.

Corollary 4.8. If µL0(`|[a+ε,b]) > n−(g,P) or µL0(`|[a+ε,b]) < − dim(P), thenthere exists at least one P-focal instant in [a+ ε, b].

Corollary 4.9. If µLP(`|[a+ε,b]) < −n−(g,P) or µLP(`|[a+ε,b]) > dim(P), thenthere exists at least one conjugate instant in [a+ ε, b].

For the following result we need to recall the definition of the space A0[t] givenin (4-4); we will denote by n+(g, t) and n−(g, t) respectively the coindex and theindex of the restriction of g to A0[t]× A0[t] and mul(t0)= dim(A0[t0]).

The estimate in Corollary 3.4 can be used to obtain results of the following type:

Corollary 4.10. If t0 ∈ ]a, b] is a conjugate instant such that either

mul(t0) > n−(g)−µL0(`|[a+ε,t0]) or µL0(`|[a+ε,t0]) <−n+(g),

then for every a′ < a there is an instant t ′ ∈ [a, t0] such that γ(t ′) is conjugate toγ(a) along γ.


Proof. Consider the Lagrangian L ′ ⊂ V given by

L ′ ={(v,w) ∈ V : Jv,w(a′)= 0

}.

If there were no instant t in [a, t0] with γ(t) conjugate to γ(a′) along γ, thenµL ′(`|[a,t0]) = dim(L ′ ∩ `(a)) = dim(L ′ ∩ `(t0)) = 0. By Corollary 3.4, it wouldthen be

µL0(`|[a,t0])= µL0(`|[a,t0])−µL ′(`|[a,t0])≤ n−mul(t0).

Using (4-6), we get a contradiction with the hypothesis of the corollary. Moreover,using (3-3), we have

mul(t0)− n = dim(`(a)∩ `(t0))− n

≤ µL0(`|[a,t0])− dim(`(a)∩ L0)+ dim(`(b)∩ L0)

= µL0(`|[a,t0])− n+mul(t0),

that is, µL0(`|[a,t0])≥ 0. This together with (4-6) concludes the proof. �

When the first conjugate point is nondegenerate, we can state a more precise result.

Corollary 4.11. Let t0 ∈ ]a, b] be the first conjugate instant along γ, and assumethat it is nondegenerate and mul(t0) > n−(g)+ n−(g, t0). Then for every a′ < athere exists and instant t ′ ∈ [a, t0] such that γ(t ′) is conjugate to γ(a′) along γ.

If g is Riemannian, then n−(g)= n−(g, t0)= 0, and the result of Corollary 4.11holds without any assumption of the multiplicity of t0.

5. Final remarks and conjectures

If the semi-Riemannian manifold (M, g) is real-analytic, then conjugate and fo-cal points do not accumulate along a geodesic, and higher order formulas for thecontribution to the Maslov index of each conjugate and focal points are available[Piccione and Tausk 2009]. In this case, the statement of all the above results canbe given in terms of the partial signatures of the conjugate and the focal points,which are a sort of generalized multiplicities.

We also observe that the nondegeneracy assumption for the conjugate and focalpoints is stable by C3-small perturbations of the metric, and generic, although aprecise genericity statement seems a little involved to prove. We conjecture that,given a differentiable manifold M and a countable set Z ⊂ T M , the set of semi-Riemannian metrics g on M having a fixed index and for which all the geodesicsγ : [0, 1] → M with γ(0) ∈ Z have only conjugate points nondegenerate and ofmultiplicity equal to 1 is generic. In this situation, the comparison results provedin this paper would have a more explicit statement in terms of number of conjugateand focal points.


A natural conjecture is also that in the case of stationary Lorentzian metrics, allgeodesics have nondegenerate conjugate points whose contribution to the Maslovindex is positive and equal to their multiplicity. This fact has been proved in thecase of left-invariant Lorentzian metrics on Lie groups of dimension less than 6[Javaloyes and Piccione 2006] and recently, using semi-Riemannian submersionsas in [Caponio et al. 2009], also for spacelike geodesics orthogonal to some time-like Killing vector field. If this conjecture were true in full generality, one wouldhave Riemannian-like comparison results also for spacelike geodesics in stationaryLorentz manifolds.

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Received October 1, 2008. Revised February 13, 2009.

MIGUEL ANGEL JAVALOYES

DEPARTAMENTO DE GEOMETRIA Y TOPOLOGIA

FACULTAD DE CIENCIAS

UNIVERSIDAD DE GRANADA

CAMPUS FUENTENUEVA S/N

18071 GRANADA

SPAIN

[email protected]

[email protected]

PAOLO PICCIONE

DEPARTAMENTO DE MATEMATICA

UNIVERSIDADE DE SAO PAULO

RUA DO MATAO, 101005508-900 SAO PAULO, SPBRAZIL

Current address:DEPARTAMENTO DE MATEMATICAS

UNIVERSIDAD DE MURCIA, CAMPUS DE ESPINARDO

30100 ESPINARDO, MURCIA

SPAIN

[email protected]://www.ime.usp.br/~piccione/








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http://dx.doi.org/10.1023/B:AGAG.0000018558.65790.db



http://dx.doi.org/10.1016/0040-9383(93)90052-W



http://dx.doi.org/10.2307/2373064






http://www.ime.usp.br/~piccione/


OUTER ACTIONS OF A DISCRETE AMENABLE GROUPON APPROXIMATELY FINITE-DIMENSIONAL FACTORS

III: THE TYPE IIIλ CASE, 0< λ < 1,ASYMMETRIZATION AND EXAMPLES

YOSHIKAZU KATAYAMA AND MASAMICHI TAKESAKI

Dedicated to the memory of Masahiro Nakamura

In this last article of the series on outer actions of a countable discreteamenable group on approximately finite-dimensional factors, we analyzeouter actions of a countable discrete free abelian group on an approximatelyfinite-dimensional factor of type IIIλ with 0 < λ < 1 and compute outerconjugacy invariants. As a byproduct, we discover the asymmetrizationtechnique for coboundary condition on a T-valued cocycle of a torsion-freeabelian group, which might have been known by group cohomologists. Asthe asymmetrization technique gives us a very handy criterion for cobound-aries, we present it here in detail.

Introduction 571. Simple examples and model construction 602. Asymmetrization 723. Universal resolution for a countable discrete abelian group 834. The characteristic cohomology group 3(Hm, L ,M,T) 915. The reduced modified HJR-sequence 1156. Concluding remark 122Acknowledgments 124References 124

Introduction

This article concludes the series [Katayama and Takesaki 2003; 2004; 2007] onthe outer conjugacy classification of outer actions of a countable discrete amenablegroup on an approximately finite-dimensional (AFD) factor, by examining outeractions of a countable discrete abelian group G on an AFD factor Rλ of type IIIλ

MSC2000: primary 46L55; secondary 46L35, 46L40, 20J06.Keywords: outer action, outer conjugacy, cohomology of group, asymmetrization.

57



58 YOSHIKAZU KATAYAMA AND MASAMICHI TAKESAKI

with 0 < λ < 1. Prior to the outer conjugacy classification theory, the cocycleconjugacy classification theory of actions of a countable discrete amenable groupon an AFD factor had been completed through the work of many mathematiciansover three decades; see [Connes 1977; 1976b; 1975; Jones 1980; Jones and Take-saki 1984; Ocneanu 1985; Katayama et al. 1998; 1997; Kawahigashi et al. 1992;Sutherland and Takesaki 1985; 1989; 1998].

Unlike the general classification program in operator algebras, the outer conju-gacy classification of a countable discrete amenable group on Rλ is almost smooth,as shown in our series of previous work; see [Katayama and Takesaki 2007].The only nonsmooth part of the classification theory stems from the classificationof subgroups N of G; for instance, the classification of subgroups of a torsion-free abelian group of higher rank is nonsmooth. See [Sutherland 1985] for theBorel parameterization of polish groups. When the modular automorphism partN = α−1(Cntr(M)) of the outer action α of G on Rλ is fixed, the set of invariantsbecomes a compact abelian group. This is a rare case in the theory of operatoralgebras. So we are encouraged to make a concrete analysis of outer conjugacyclasses of a countable discrete amenable group. Of course, without having a con-crete data on the group G involved, we cannot make a fine analysis. So we takea countable discrete free abelian group G and study its outer actions on Rλ andidentify the invariants completely. The justification of this restriction rests on thefact that all outer actions of a countable discrete abelian group A can be viewed asouter actions of G by pulling back the outer action via the quotient map G→ A.Thanks to all hard analytic work on the cocycle conjugacy classification in thepast, as cited in the references, our work is very algebraic and indeed done bycohomological computations.

We will begin by relating the discrete core of Rλ and the core of an AFD factorR1 of type III1. This analysis will give us a simple model construction with giveninvariants, which is presented here in Section 1. We first study single automor-phisms and a pair of commuting automorphisms of Rλ. Then we will work onthe asymmetrization of a cocycle of a countable discrete abelian group; this willprovide a powerful tool for analysis of the third cohomology group H3(G,T).The general theory of group cohomology is available to us today; for examplesee [Brown 1994]. But, since we will need to work with individual cocycles toanalyze outer actions, we will need a tool to work with a cocycle directly beyondthe computation of the cohomology group. For example, we have to identify whichdata of a given cocycle contributes to the modular automorphism part of the actionin question. Thus we will work on the cohomology group based on a very primitivemethod of chasing cocycles, through which we discover the asymmetrization tech-nique that provides us a handy criterion for the coboundary condition on a cocycleof a torsion-free abelian group. In [Katayama and Takesaki 2003; 2004; 2007], we

OUTER ACTIONS OF A DISCRETE AMENABLE GROUP, III 59

studied the outer conjugacy classification of countable discrete amenable groupouter actions by a resolution of the relevant third cocycle. In the abelian case, weshowed that there is a universal resolution group that takes care of all third cocyclesat once, which simplifies greatly the investigation of outer actions of a countablediscrete abelian group. The reduced modified HJR-sequence will provide us a toolto chase the cocycles, along with the asymmetrization technique. The first stepof studying outer actions of a countable discrete abelian group G on a factor M

of type IIIλ with 0 < λ < 1 is to find a countable discrete amenable group Hand a surjective homomorphism πG : H → G such that the pull back π∗G(c) is acoboundary; this process is called the resolution of a cocycle c ∈ Z3(G,T). Thenthe outer action α is identified with a lifting s∗H (α) of an action α of H through across-section sH :G→ H of the homomorphism πG . Luckily, a countable discreteabelian group G admits a universal resolution {H, πG}, a group H and a surjectivehomomorphism πG :H→G such that π∗G(H

3(H,T))={1}. We construct the groupH via a relatively simple process from a countable discrete free abelian group G.This makes it possible to reduce the study of an outer action α of G to that of anaction α of H . Now, the action α of H does not lift to the discrete core Md ifmod(α) 6= 1. So we construct a central extension Hm of H by

0→ Zn→zn

0 // Hm // H → 1

and work with the characteristic cohomology group 3(Hm, L ,M,T), where thenormal subgroup L is the inverse image L = π−1

G (N ) with N = α−1(Cntr(M)).Thus we are going to investigate the reduced modified HJR-sequence

H2(H,T)Res // 3(Hm, L ,M,T)

δ //

π∗m��

Houtm,s(G, N ,T)

Inf //

∂Gm��

H3(H,T)

H2(H,T)res // 3(H,M,T)

δHJR // H3(G,T)π∗G // H3(H,T).

Here s is a fixed cross-section of the quotient map G → Q = G/N . The groupsappearing on the exact sequences above are all compact abelian groups and areindeed computable as shown in this paper.

We cite [Brown 1994; Eilenberg and Mac Lane 1947; Mac Lane and Whitehead1950; Huebschmann 1981; Jones 1980; Ratcliffe 1980] for the general cohomologytheory of abstract groups and [Sutherland 1980] for the cohomology theory relatedto von Neumann algebras. See [Takesaki 1979; 2003a; 2003b] for the generaltheory of von Neumann algebras. For information about the discrete core of afactor of type IIIλ, see [Connes 1973; 1974; Connes and Takesaki 1977; Falconeand Takesaki 1999; 2001].


1. Simple examples and model construction

Factors of type IIIλ and type III1, and their cores. We begin with the followingfolk theorem in the structure theory of factors of type III.

Theorem 1.1. Let {M0,1, τ, θ} be a factor of type II∞ equipped with faithful semi-finite normal trace τ and trace scaling automorphism θ by λ with 0 < λ < 1, thatis, τ ◦ θ = λτ . Let M=Mθ

0,1 be the fixed point subalgebra of M0,1 by θ .

(i) The von Neumann algebra M is a factor of type IIIλ.(ii) The triplet {M0,1, τ, θ} is conjugate to the discrete core of M.

(iii) For an automorphism α ∈ Aut(M0,1),

α(M)=M is equivalent to α ◦ θ = θ ◦ α.

(iv) Let Aut(M0,1,M) be the group of automorphisms of Aut(M0,1) leaving M

globally invariant. Then we have the exact sequence

0→ Zn→θn

// Aut(M0,1,M)α→α|M // Aut(M)→ 1.

(v) The subgroup {θn: n ∈ Z} is the Galois group of the pair {M0,1,M} in the

sense that {θn: n ∈ Z} = {a ∈ Aut(M0,1) : α(x)= x, x ∈M}.

(vi) If α∈Aut(M0,1,M), then the modulus modM0,1(α) as a member of Aut(M0,1)

gives the modulus modM(α) of the restriction α|M ∈ Aut(M) as

modM(α)= πT ′(modM0,1(α)) ∈ R/T ′Z,

where T ′ =− log λ, T = 2π/T ′ and πT ′ : s ∈ R 7→ sT ′ = s+ T ′Z ∈ R/T ′Z.

Proof. Statements (i) and (ii) are known in the general structure theory of a factorof type III; see [Takesaki 2003a, Chapter XII, Sections 2 and 6].

We prove statement (v) first. Letψ be a generalized trace of M, that is, a faithfulsemifinite normal weight on M such that ψ(1) = +∞ and σψT = id. Then thecovariant system {M0,1, θ} is conjugate to the dual system {Moσψ R/T Z,Z, σψ }.Thus we may identify them, so that M0,1 admits a periodic one parameter unitarygroup {uψ(s) : s ∈ R} with

uψ(T )= 1, θ(uψ(s))= λisuψ(s), and σψs = Ad(uψ(s))|M for s ∈ R.

Furthermore, the one parameter unitary group {uψ(s) : s ∈R} together with U(M)

generates the normalizer U0(M) = {v ∈ U(M0,1) : vMv∗ =M}, giving the semi-direct product decomposition U0=U(M)oσψ R/T Z. Suppose that α ∈Aut(M0,1)

leaves M pointwise fixed. We then show that x ∈M and uψ(s)∗α(uψ(s)) for s ∈R

commute by calculating

xuψ(s)∗α(uψ(s))= uψ(s)∗uψ(s)xuψ(s)∗α(uψ(s))= uψ(s)∗σψs (x)α(uψ(s))

= uψ(s)∗α(σψs (x)uψ(s))= uψ(s)∗α(uψ(s)x)= uψ(s)∗α(uψ(s))x,


so that uψ(s)∗α(uψ(s)) ∈M0,1 ∩M′ = C. Hence there exists a scalar µ(s) ∈ T

such that α(uψ(s))= µ(s)uψ(s) for s ∈ R. Since uψ(T )= 1, we have µ(T )= 1.Since µ(s + t) = µ(s)µ(t) for s, t ∈ R, we have µ(s) = λins for s ∈ R and somen ∈ Z. Since M together with {uψ(s) : s ∈R} generate the whole algebra M0,1, weconclude that α = θn . This shows (v).

We next show (iii). Suppose that α ∈ Aut(M0,1) leaves M globally invariant.Let β = αM = α|M be the automorphism of M obtained as the restriction of αto M. Then the uniqueness of a generalized trace on M gives a scalar s ∈ R and aunitary v ∈ U(M) such that e−sψ = ψ ◦ (Ad(v) ◦ β). This means that

mod(β)=mod(Ad(v) ◦ β)= sT ′ = s+ T ′Z ∈ R/T ′Z,

and that σψ and Ad(v) ◦ β commute. Hence it is possible to extend Ad(v) ◦ β tothe automorphism γ ∈ Aut{M0,1} such that

γ(uψ(t))= uψ(t) for t ∈ R and γ(x)= Ad(v) ◦ β(x) for x ∈M.

Now comparing α and γ on M, we find γ(x) = Ad(v) ◦ β(x) = Ad(v) ◦ α(x) forx ∈M. From (v) it follows that α is of the form α= θn

◦Ad(v∗)◦γ for some n ∈Z.Since θ commutes with both γ and Ad(v), α and θ commute. Hence α(M)=M

implies α ◦ θ = θ ◦ α. The reverse implication is trivial. This proves part (iii).Part (iv) follows from (iii) and (v).Let {M,R, τ, θ} be the noncommutative flow of weights on M, so that the co-

variant system {M0,1,Z, θ} is identified with {M∨{ψ isρ(−s) : s ∈R}, θT ′}, whereρ(s) for s ∈ R is the one-parameter unitary group generating the center C of M

such that ρ(T )= ψ iT .To prove (vi), fix a member α ∈ Aut(M0,1,M) and let m(α) = mod(α) ∈ R so

that τ ◦α= e−m(α)τ . Consider the crossed product M=M0,1oθZ∼=M⊗L(`2(Z))

and the generalized trace ϕ = τ ◦ E on M given by

ϕ(x)= τ ◦ E(x)= τ(∫

R/T Z

θs(x) ds)

for x ∈ M+.

With U ∈ U(M) the unitary corresponding to the crossed product M0,1 oθ Z, weextend α to α ∈ Aut(M) by α(x) = α(x) for x ∈M0,1 and α(U ) = U . Then wehave for each x ∈ M+

ϕ ◦ α(x)= τ(∫

R/T Z

θs(α(x)) ds)= τ

(α(∫

R/T Z

θs(x) ds))

= e−m(α)τ(∫

R/T Z

θs(x)ds)= e−m(α)ϕ(x).


Hence we get

(1-1) mod(α)= [m(α)]T ′ =m(α)+ T ′Z ∈ R/T ′Z.

Since the covariant systems {M, α} and {M, α} are cocycle conjugate, we havemod(α)=mod(α). This completes the proof. ♥

From now on we denote by R0 an approximately finite-dimensional factor oftype II1.

A factor M1 of type III1 generates a one-parameter family {Mλ : 0< λ ≤ 1} offactors of type IIIλ, who share the same discrete core M0,1. So let M1 be a factor oftype III1, and let {M0,1, θs, s ∈ R} be the noncommutative flow of weights on M1,that is, M0,1 is a factor of type II∞ equipped with a trace-scaling one-parameterautomorphism group {θs : s ∈ R} and a faithful semifinite normal trace τ such thatM1 =Mθ

0,1 and τ ◦ θs = e−sτ for s ∈ R. The following is a folk theorem in thestructure theory of type III.

Theorem 1.2. In the above context, fixing T ′ > 0, set λ = e−T ′ and T = 2π/T ′.Let Mλ be the fixed point subalgebra M

θT ′

0,1 of M0,1 under the automorphism θT ′ .

(i) The subalgebra Mλ ⊂ M0,1 is a factor of type IIIλ, whose discrete core isconjugate to the pair {M0,1, θT ′}.

(ii) The triplet {M0,1,Mλ, θT ′} is a Galois triplet in that

Gal(M0,1/Mλ)= {θnT ′ : n ∈ Z},

where Gal(M/N) = {α ∈ Aut(M) : α|N = id} for any pair N ⊂ M of vonNeumann algebras. We have the exact sequence

1→ {θnT ′ : n ∈ Z} // Aut(Mλ)m // Aut(Mλ)→ 1,

andAut(Mλ)m = {α ∈ Aut(M0,1) : α(Mλ)=Mλ}

= {α ∈ Aut(M0,1) : α ◦ θT ′ = θT ′ ◦ α}.

(iii) Another pair {Mλ,M1} forms a Galois pair

Gal(Mλ/M1)= {θsT ′: sT ′ = s+ T ′Z ∈ R/T ′Z, s ∈ R},

that is, an α ∈ Aut(Mλ) is of the form α = θsT ′for some sT ′ ∈ R/T ′Z if and

only if α(x)= x for x ∈M1.

(iv) The modulus of θsT ′∈ Aut(Mλ) is precisely −sT ′ ∈ R/T ′Z itself , that is,

mod(θsT ′)=−sT ′ ∈ R/T ′Z.

If any of Mλ,M1 and M0,1 is approximately finite-dimensional, then all others areapproximately finite-dimensional, and the following statements hold:


(v) If α ∈ Aut(Mλ) has aperiodic modulus m = mod(α), that is, if km 6= 0 forevery nonzero integer k ∈ Z or equivalently if {mod(α)}T ′/T ′ 6∈ Q, then α iscocycle conjugate to θ−m.

(vi) If an automorphism α ∈ Aut(Mλ) has trivial asymptotic outer period, thatis, pa(α) = 0, then its cocycle conjugacy class is determined by its modulusm = mod(α) ∈ R/T ′Z. In fact, the automorphism α is cocycle conjugate tothe automorphism θ−m ⊗ σ0 on Mλ

∼=Mλ ⊗R0, where σ0 ∈ Aut(R0) is anyaperiodic automorphism of the approximately finite-dimensional factor R0. Ifm 6= 0, then θm ∼ θm⊗ σ0.

Proof. We prove statements (v) and (vi). Choose an automorphism α ∈ Aut(Mλ)

such that m = mod(α) is aperiodic. Let R0 be an AFD factor of type II1 realizedas the infinite tensor product of two by two matrix algebras

R0 =∏⊗

n∈Z

{Mn, τn}

relative to the normalized traces τn = Tr /2 on Mn = M(2,C). Let σ0 be theBernoulli shift automorphism of R0, that is, the automorphism determined by

σ0

(∏⊗

n∈Z

xn

)=

∏⊗

n∈Z

xn+1.

Then by the grand theorem of Connes [1975] (also [Takesaki 2003b, page 267])α and α⊗ σ0 are cocycle conjugate under the identification of Mλ and Mλ ⊗R0

because the asymptotic outer period pa(α) of α is zero, that is, pa(α) = 0. Thesame is true for θm, that is, θm ∼c θm⊗ σ0, where ∼c means the outer conjugacy.Since mod(α1 ⊗ α2) = mod(α1)+mod(α2) on Mλ ⊗Mλ

∼= Mλ, we have α ∼c

α⊗ σ0 ∼c α⊗ θm⊗ θ−m ∼c σ0⊗ θ−m ∼c θ−m. This proves statement (v).To prove (vi), suppose that p∈N is the period of m∈R/T ′Z, that is, the smallest

nonnegative integer with pm= 0. We assume that p 6= 0. Let {e j,k : 1≤ j, k ≤ p}be the standard matrix units of the p × p matrix algebra M(p;C), and for eachn ∈ N set Mn =M(p;C). Also consider the diagonal unitary

un =p∑

i=1exp(2π i((i − 1)/p))ei,i ∈ U(p;C)⊂ Mn

of order p, that is, u pn = 1. Now we identify the AFD factor R0 with the infinite

tensor product

R0 =∏⊗

n∈N

{Mn, τn}, where τn =1n Tr,

and letσp =

∏⊗

n∈N

Ad(un) ∈ Aut(R0) ∈ Aut(R0).


Then the automorphism σp has the properties

σ kp 6∈ Int(R0) for k = 1, . . . , p− 1, and σ p

p = id,

θm ∼c θm⊗ σp on Mλ∼=Mλ⊗R0,

θm⊗ θ−m ∼c id⊗ id⊗σp on Mλ⊗Mλ∼=Mλ⊗Mλ⊗R0,

σ0⊗ σp ∼c σ0 on R0⊗R0 ∼= R0.

If α ∈ Aut(Mλ) has the trivial asymptotic outer period pa(α) = 0, then the auto-morphism α has the properties

α ∼c σp⊗α on Mλ∼= R0⊗Mλ,

θm⊗α ∼c id⊗σ0 on Mλ⊗Mλ∼=Mλ⊗R0,

θ−m⊗ σ0 ∼c θ−m⊗ θm⊗α ∼c σp⊗α ∼c α

under the isomorphisms Mλ ⊗ R0 ∼= Mλ ⊗Mλ ⊗Mλ∼= R0 ⊗Mλ

∼= Mλ. Thiscompletes the proof. ♥

Thus if mod(α) is aperiodic, or pa(α) = 0, then the grand theorem of Connes[Connes 1975], or see [Takesaki 2003b, page 270], identifies the cocycle conjugacyclass of α ∈ Aut(Mλ). But if mod(α) has nontrivial period, and p1 = pa(α) 6= 0,then the cocycle conjugacy class of α involves algebraic invariants. For example,one has to consider the extension of α to the discrete core Mλ,d on which α alonecannot act. In fact, one has to consider a larger group Z2 than the integer group Z.

Invariants for single automorphisms. We consider a single automorphism of afactor M of type IIIλ, which can be viewed as an action of the integer additivegroup Z. As the integer group Z appears in many different roles, we denote it hereby G = Z. Let a1 be the generator of the group G, so that G = Za1. Sometimeswe view G as a multiplicative group, in which case G becomes G = {ak

1 : k ∈ Z}.Since H2(G,T) = H3(G,T) = {1}, that is, the integer group is cohomologicallytrivial, there is no distinction between the cocycle conjugacy problem and the outerconjugacy problem of actions of G. Namely, an outer action α of G always comesfrom an action α of G, and outer conjugacy of the outer action α of G is the same asthe cocycle conjugacy of the action α of G. Hence the obstruction Ob(α) of α andthe characteristic invariant χ(α) of α are handily identified. The same is true forthe modular obstruction Obm(α) and the modular characteristic invariant χm(α).

Since the single automorphism cocycle conjugacy classification wasn’t handledproperly in [Katayama et al. 1998; 1997], and more importantly because the pre-sentation of a single automorphism on a factor of type IIIλ in [Takesaki 2003b]contains a minor mistake, we present it here in some detail.


In the case that the modulus m = mod(α) is aperiodic, the last theorem takescare of the cocycle conjugacy of α, that is, it must be cocycle conjugate to θ−m.So we handle only the case that {mod(α)}T ′ is rational multiple of T ′.

Suppose α−1(Cntr(M))= Zb1 and b1 = p1a1, with p1 ∈ N.Choose a pair p1, q1 ∈ N of positive integers with q1 < p1 such that

m= (q1/p1)T ′+ T ′Z ∈ R/T ′Z for 0≤ q1 < p1.

Form a group extension

(1-2)Gm = {(g, s) ∈ G×R : gm= sT ′ = s+ T ′Z ∈ R/T ′Z}

0→ Zk→(0,kT ′) // Gm

pr1 // G→ 0.

Set

(1-3)z0 = (0, T ′), z1 = (a1, {m}T ′),

b1 = p1z1− q1z0, N = Zb1, Qm = Gm/N .

The group Gm is equipped with a distinguished homomorphism km = pr2 to R:

(1-4) km(g, s)= s ∈ R for (g, s) ∈ Gm.

Let πQ : g ∈ Gm 7→ g ∈ Qm be the quotient map and further set

(1-5) D1 = gcd(p1, q1), and r1 = p1/D1, s1 = q1/D1,

and find a pair u1, v1 ∈ Z of integers such that

1= r1u1− s1v1, or equivalently D1 = p1u1− q1v1,

which can be done through the Euclid algorithm. In the event that q1 = 0, themodulus m is trivial, that is, m= 0 and Gm = G⊕Z.

Theorem 1.3 (invariants for a single automorphism with periodic modulus). SetD1= gcd(p1, q1). If p1 and q1 are both nonzero, we have the following statements:

(i) The pair {z0, z1} is a free basis of Gm, so that every element g ∈Gm is writtenuniquely in the form g = e0(g)z0+ e1(g)z1.

(ii) The group Gm admits another free basis {w0, w1} such that b1=D1w1. There-fore N = D1Zw1 and

Qm = Zw0⊕Zw1,

D1w1 = 0 in Qm ∼= Z⊕ZD1,

where the dotted notations indicate their images in the quotient group Qm.


(iii) The character group Qm of Qm and the characteristic cohomology group3(Gm, N ,T) are identified under the correspondence

(1-6) λχ (nb1; g)= χ(πQ(g))n for g ∈ Gm and χ ∈ Qm.

(iv) The character group Qm is given by the exact sequence

0→ Z2 // R⊕ ( 1D1

Z)exp(2π i ·) // T⊕ZD1 = Qm→ 0,

which describes the characteristic cohomology group 3(Gm, N ,T) as

(1-7) 3(Gm, N ,T)∼= T⊕ZD1 .

If χ(z0) is a root of unity, then the outer period po(α) of α is given as the productp1so with so ∈ Z+ the smallest nonnegative integer s ∈ Z+ such that 1= χ(z0)

s . Ifχ(z0) is not a root of unity, then the corresponding automorphism α is aperiodic,that is, po(α)= 0.

Proof. (i) Since pr1(z1) = a1 and G is a free abelian group, the exact sequence(1-2) splits along with the cross-section: m ∈ G 7→ mz1 ∈ Gm.

(ii) We set w0 = u1z0− v1z1 and w1 =−s1z0+ r1z1. Since z0 = r1w0+ v1w1 andz1 = s1w0+ u1w1, the pair {w0, w1} is a free basis of Gm such that

Gm = Zw0+Zw1, b1 = D1w1, N = D1Zw1, Qm = Gm/N = Zw0⊕Zw1,

as we wanted.

(iii) Since H2(N ,T) = {1}, the second cocycle part of a characteristic cocycle inZ(Gm, N ,T) is taken to be trivial, so that the λ-part vanishes on N and thereforeit is a character of Gm that vanishes on N and factors through the quotient mapπQ :Gm→Qm. Thus it is of the form λ(b1; g)=χ(πQ(g)) for g∈Gm and χ ∈ Qm.

(iv) It follows from (ii) that the character group Qm is parameterized by R⊕( 1D1

Z):

χx,y(g)= exp(2π i(x f0(g)+ y f1(g))) for g = f0(g)w0+ f1(g)w1 ∈ Gm,

with (x, y) ∈ R⊕ ( 1D1

Z). This gives the exact sequence

0→ Z2 // R⊕ ( 1D1

Z)(x,y)→χx,y // Qm = T⊕ZD1 → 0. ♥

Model construction. Suppose G is a fixed countable discrete amenable group andlet {H, πG} be a universal resolution group of the third cocycles of G, that is,πG : H → G is a surjective homomorphism such that π∗G(Z

3(G,T)) ⊂ B3(H,T).We require H to be a countable discrete amenable group. Let M = Ker(πG). Fixa normal subgroup N of G, and set L = π−1

G (N ). With a fixed invariant homo-morphism m ∈ HomG(N ,R/T ′Z) such that Ker(m) ⊃ N , we use the abbreviated


notation m for m ◦ πG and form a group extension Hm via

0→ Z // Hmπm // H → 1,

where Hm={(g, s)∈H×R :m(g)= sT ′= s+T ′Z∈R/T ′Z}, with πm(g, s)=g∈Hand k(g, s)= s ∈ R for (g, s) ∈ Hm. We get the reduced modified HJR-sequence

· · · // H2(H,T)Res // 3(Hm, L ,M,T)

δ // Houtm,s(G, N ,T)→ 1.

Thus every modular obstruction cocycle (c, ν) ∈ Zoutm,s(G, N ,T) is of the form

(c, ν)≡ δ(λ, µ) mod Boutm,s(G, N ,T).

Consequently the construction of an outer action α of G on an AFD factor Mλ oftype IIIλ with Obm(α)= ([c], ν)∈Hout

m,s(G, N ,T) is reduced to the construction ofan action αλ,µ of Hm such that

(αλ,µ)−1(Int(Mλ))⊃ M, χ(αλ,µ)= [λ,µ] ∈3(Hm, L ,M,T),

(αλ,µ)−1(Cnt(Mλ))= L , mod (αλ,µg )=m(πG(g)) for g ∈ Hm.

So fix a set of invariants (λ, µ)∈Z(Hm, L ,M,T) and m∈HomG(G,R/T ′Z) suchthat Ker(m)⊃ N . We are going to construct the model action αλ,µ of Hm:

Step I. Let X be a countable but infinite set on which Hm acts freely from the left.In the case that Hm is an infinite group, we take X to be Hm itself and let Hm acton it by left multiplication. So the infinite set X is only needed when Hm is a finitegroup, in which case X can be taken to be the product set X = Hm ×N and Hm

acts on the first component by left multiplication. Let {Mx , x ∈ X} be the set of2× 2 matrix algebras M(2,C) indexed by elements x ∈ X .

Step II. We form the infinite tensor product R0 =∏⊗

x∈X {Mx , τx} relative to thenormalized trace

τx

(a11 a12

a21 a22

)=

12(a11+ a22).

Then we let σ 0 be the Bernoulli action of Hm on R0 that is determined by

σ 0g

(∏⊗

x∈X

ax

)=

∏⊗

x∈X

agx .

Step III. Form the twisted partial crossed product of R0 by N relative to the secondcocycle µ ∈ Z2(N ,T) and the action σ 0, that is, M0 = R0 oσ 0,µ N . Then let{U (m) :m ∈ N } be the projective unitary representation of N to M0 correspondingto the twisted crossed product, so that

U (g)U (h)= µ(g; h)U (gh) for g, h ∈ N ,

U (g)aU (g)∗ = σ 0g (a) for a ∈ R0, g ∈ N .


Let σ λ,µ be the action of Hm on M0 determined by

σ λ,µg (U (m))= λ(gmg−1; g)U (gmg−1) for m ∈ N and g ∈ Hm,

σ λ,µg (a)= σ 0g (a) for a ∈ R0 and g ∈ Hm.

Step IV. Let M0,1 be the AFD factor of type II∞ equipped with trace-scaling oneparameter automorphism group {θs : s ∈ R} and set R0,1 =M0,1⊗M0. We thenset the action αλ,µ by αλ,µg = θm(g)⊗σ

λ,µg on R0,1 for g ∈ Hm. Set R= (R0,1)

αz0 .The automorphism αz0 = θT ′ ⊗ σ

λ,µz0 scales the trace τ by λ = e−T ′ , so the von

Neumann algebra R is an AFD factor of type IIIλ. Finally we define the actionαλ,µ by αλ,µg = α

λ,µg |R for g ∈ H ; this makes sense because αz0 acts trivially on R.

Theorem 1.4 (model action).(i) The action α = αλ,µ constructed above has the invariants

N = α−1(Cnt(Rλ)),mod (αg)=m(g) for g ∈ H,

χ(α)= [λ,µ] ∈3(Hm, L ,M,T),

να(g)= [T Log(λ(g; z0))/2π ]T ∈ R/T Z for g ∈ N .

(ii) Let sH : G → H be a cross-section of the homomorphism πG : H → G.Then the outer action αλ,µsH of G has associated modular obstruction given byδ([λ,µ])= [cλ,µ, νλ] ∈ Hout

m,s(G, N ,T).

The construction of (i) and (ii) exhausts all outer actions of G on the approxi-mately finite-dimensional factor R of type IIIλ, up to outer conjugacy.

Proof. (i) Let α denote the action αλ,µ of Hm on R0,1. Since R is the fixed pointsubalgebra of R0,1 under the automorphism αz0 , the restriction α = α|R of α to R

factors through the quotient group H = Hm/(Zz0). Hence α is indeed an actionof H . Since R0,1 is a factor of type II∞ and

τ ◦ αz0 = τ ◦ θm(z0) = e−m(z0)τ = e−T ′τ = λτ,

the fixed point subalgebra R is a factor of type IIIλ and the pair {R0,1, αz0} is thediscrete core of the factor R. Since R0,1 is AFD, R is as well by the grand theoremof Connes [1976a]. As z0 is a central element of Hm, α(Hm) leaves R globallyinvariant and hence its restriction to R makes sense. The inner part α(N ), which isgiven by the projective representation {U (g) : g ∈ N }, leaves R globally invariant,that is, U (g) for g ∈ N normalizes R; thus we have the inclusion U (N )⊂ U0(R).Hence N = α−1(Cnt(R)). As in (1-1), we have mod(αh) = m(h) for h ∈ H . Ifg, g1, g2 ∈ N and h ∈ H , then

λ(g; h)=U∗(g)αh(U (h−1gh)), U (g1)U (g2)= µ(g1; g2)U (g1g2),

να(g)= ∂αz0(U (g))=U (g)∗αz0(U (g))= λ(g; z0).


Hence χ(α) = [λ,µ] ∈ 3(Hm, L ,M,T) as required. Finally viewing να as ahomomorphism of N into R/T Z, we get να ∈ HomG(N ,R/T Z) as stated.

(ii) The assertion follows from the construction of αλ,µ. ♥

Actions and outer actions of two commuting automorphisms on an AFD factor RRR

of type IIIλ. In this case, we have to consider the free abelian group G=Z2 of ranktwo and its extension Gm∼=Z3 relative to a homomorphism m :G→R/T ′Z. We fixa subgroup N of G, which is going to represent the inverse image α−1(Cnt(Mλ))

of the extended modular automorphism group. We assume that N is in the diagonalform, that is, with a free basis {a1, a2} of G, the subgroup N is of the form N =p1Za1 + p2Za2. Of course, one can choose p1 and p2 so that 0 ≤ p1 ≤ p2 andp1 divides p2, but to go beyond the finite rank case, we don’t assume that p1 is adivisor of p2, which might make the matter slightly more involved. In the case thatG =Z2, we have H3(G,T)= {1}, so every outer action of G comes from an actionof G. Since H2(G,T) ∼= T 6= {1}, the outer conjugacy class of an action is biggerthan the cocycle conjugacy class. To go further, we recall the reduced modifiedHJR-exact sequence from [Katayama and Takesaki 2007, Theorem 3.11 page 116]:

H2(G,T)ResQm // 3(Gm, N ,T)

δQm // Houtα,s(G, N ,T)

InfQm // H3(G,T)= {1},

where Qm = Gm/N . Here since H3(G,T) = {1}, we don’t have to consider theresolution group H and its subgroup M . To identify the subgroup N ⊂ G as asubgroup of Gm, we need a little care. First, set

(1-8)

z0 = (0, T ′) ∈ Gm,

z1 = (a1, q1T ′/p1) ∈ Gm, z2 = (a2, q2T ′/p2),

b1 = (p1a1, 0)= p1z1− q1z0 ∈ Gm,

b2 = (p2a2, 0)= p2z2− q2z0 ∈ Gm,

N = Zb1+Zb2 ⊂ Gm = Zz0+Zz1+Zz2,

Qm = Gm/N .

This gives the following coordinate system in Gm and N :

(1-9)g = e1,N (g)b1+ e2,N (g)b2 ∈ N , that is, ei,N (g)=

ei (g)pi

for i = 1, 2,

h = e0(h)z0+ e1(h)z1+ e2(h)z2 ∈ Gm.

Theorem 1.5 (invariant). Define Z and B by

(1-10) Z={b = {b(i, j) : i = 1, 2, j = 0, 1, 2} ∈ R6

:

p j b(i, j)− q j b(i, 0) ∈ Z, i = 1, 2, j = 1, 2},

B={b ∈ Z : b(i, 0), b(i, i) ∈ Z, i = 1, 2, p2b(1, 2)+ p1b(2, 1) ∈ gcd(p1, p2)Z

},


and to each b ∈ Z associate a cochain (λb, µb) ∈ Z(Gm, N ,T) by

(1-11)λb(g; h)= exp

(2π i

( ∑i=1,2; j=0,1,2

b(i, j)ei,N (g)e j (h))),

µb(g1; g2)= 1 for g, g1, g2 ∈ N and h ∈ Gm.

Then the cochain (λb, µb) is a characteristic cocycle (λb, 1) ∈ Z(Gm, N ,T). Themodular obstruction cocycle (cb, νb)= δ(λb, 1) ∈ Zout

m,s(G, N ,T) corresponding to(λb, 1) takes the form

(1-12)

cb(g1; g2; g3)= λb(nN(g2; g3); s(g3)) (where g1, g2, g3 ∈ Qm)

= exp(

2π i( ∑

i=1,2j=0,1,2

b(i, j)ηpi ([ei (g2)]pi ; [ei (g3)]pi ){e j (g1)}p j

pi

)),

νb(g)=[

T∑

i=1,2

b(i, 0)ei,N (g)]

T∈ R/T Z for g ∈ N ,

where for the notations ηpi and nN we refer to definitions (3-8) and (3-14), andfurthermore {e0(g1)}p0 = e0(g1) ∈ Z for g1 ∈ Qm. The (i, j)-components Z(i, j)and B(i, j) of Z and B give more precise information about the cocycles:

(i) For i = 1, 2, we have

(1-13)Zb(i, i)= {z = (x, u) ∈ R2

: pi x − qi u ∈ Z},

Bb(i, i)= Z⊕Z.

The bicharacter λi,iz on N ×Gm determined by

(1-14) λi,iz (g; h)= exp(2π i(xei,N (g)ei (h)+ uei,N (g)e0(h)))

for each pair g∈N and h∈Gm gives a characteristic cocycle of Z(Gm, N ,T).It is a coboundary if and only if z is in Bb(i, i). The corresponding coho-mology class [λi,i

z ] ∈3b(i, i) has the parameterization

(1-15)[λi,i

z ] ∈3(i, i)∼ ([pi x − qi u]gcd(pi ,qi ), [−vi x + ui u]Z)

∈ Zgcd(pi ,qi )⊕ (R/Z),

where the integers ui and vi are determined by pi ui − qivi = gcd(pi , qi )

for i = 1, 2 through the Euclid algorithm. For the same i , the associatedmodular obstruction cohomology class ([ci,i

z , νi,iz ]) ∈ Hout

m,s(i, i) correspondsto the class:

([pi x − qi u]gcd(pi ,qi ), [−vi x + ui u]Z) ∈ Zgcd(pi ,qi )⊕ (R/Z),

νi,iz (g)= [T uei,N (g)]T ∈ R/T Z.


(ii) With (i, j)= (1, 2),

(1-16)

Zb(i, j)={(x, u, y, v) ∈ R4

: p j x − q j u ∈ Z, pi y− qiv ∈ Z};

Bb(i, j)={(x, u, y, v) ∈ Zb(i, j) : p j x + pi y ∈ gcd(pi , p j )Z,

u, v ∈ Z}.

For each element z= (x, u, y, v)∈Zb(i, j), the corresponding bicharacter λz

on N ×Gm determined by

(1-17)λi, j

z (g; h)= exp(2π i(xei,N (g)e j (h)+ ye j,N (g)ei (h))

)× exp

(2π i(uei,N (g)e0(h)+ ve j,N (g)e0(h))

),

for each pair g∈ N and h∈Gm is a characteristic cocycle in Z(Hm, L ,M,T).It is a coboundary if and only if z belongs to Bb(i, j). The cohomology class[λ

i, jz ] ∈3b(i, j) of λz corresponds to the parameter class

(1-18)

[z] =

[mi, j (xr j,i + yri, j )− ni, j (us j,i + vsi, j )]Z[yi, j (xr j,i + yri, j )+ xi, j (us j,i + vsi, j )]Z

[−uwi, j + vw j,i ]Z

∈( 1

D(i, j)Z)/

Z

R/Z

R/Z

,where the integers D(i, j), . . . , wi, j are those such that

(1-19)

D(i, j)= gcd(pi , p j , qi , q j ),

Di, j = gcd(pi , p j ), Ei, j = gcd(qi , q j ),

ri, j = pi/Di, j , r j,i = p j/Di, j

si, j = qi/Ei, j , s j,i = q j/Ei, j ,

mi, j = Di, j/D(i, j), ni, j = Ei, j/D(i, j),

qiwi, j + q jw j,i = Ei, j , xi, j Di, j + yi, j Ei, j = D(i, j).

The associated modular obstruction class ([ci, jz ], ν

i, jz ) ∈ Hout

m,s(i, j) corre-sponds to the pair of classes

[z] ∈

( 1

D(i, j)Z)/

Z

R/Z

R/Z

,νi, j

z (g)=[T(

ue1(g)

p1+ v

e2(g)p2

)]T∈ R/T Z for g ∈ N .

The proof of this special case is not much simpler than the general case, so we willdiscuss later in the general free abelian group case; see Theorem 4.2.


2. Asymmetrization

Set the notations X = Zn+1 = Z/(n + 1)Z and X1 = X\{1}. The signature of apermutation σ is the sign of the product: sign(σ )= sign

{∏i< j (σ ( j)−σ(i))

}. Let

S be the cyclic permutation

(2-1) S = (2, 3, . . . , n, n+ 1, 1) ∈5(X),

whose signature is given by

(2-2) sign(S)= (−1)n.

Each element σ ∈5(X1) is identified with an element of 5(X) so that

σ = (1, σ (2), σ (3), . . . , σ (n), σ (n+ 1)) ∈5(X).

This identification of an element of 5(X1) with the corresponding element of5(X) preserves the signature of σ . Then the total permutation group 5(X) isthe disjoint union of the translations {Sk5(X1) : 0≤ k ≤ n}, that is,

(2-3) 5(X)=n⊔

k=0

Sk5(X1).

Definition 2.1. The asymmetrization AS ξ of ξ ∈ Cn(G, A) is defined by

(2-4) (AS ξ)(g1, g2, . . . , gn)=∑

σ∈5(Zn)

sign(σ )ξ(gσ(1), gσ(2), . . . , gσ(n)).

Define πk : Gn+1→ Gn by

(2-5) πk(g1, g2, . . . , gn, gn+1)

=

(g2, g3, . . . , gn, gn+1) for k = 0,(g1, . . . , gk−1, gk gk+1, gk+2, . . . , gn+1) for 1≤ k ≤ n,(g1, g2, . . . , gn) for k = n+ 1.

The boundary operation d ∈ Hom(Z(Gn+1),Z(Gn)) is then given by

(2-6) d =n+1∑k=0

(−1)k ◦ πk,

∂ξ = d∗ξ for ξ ∈Cn+1(G,T). We view the asymmetrization AS also as an elementof End(Z(Gn)) determined by

AS(g1, g2, . . . , gn)=∑

σ∈5(Zn)

sign(σ )(gσ(1), gσ(2), . . . , gσ(n)).


Lemma 2.2. The asymmetrization and the boundary operation are related by

AS ◦ d = 0 in Hom(Z(Gn+1),Z(Gn)).

Proof. Define Q ∈ Hom(Z(Gn+1),Z(Gn)) and R ∈ Hom(Z(Gn+1),Z(Gn)) by

Q =∑

σ∈5(X1)

n+1∑j=1

(sign(S j−1σ)π0S j−1σ + (−1)n+1 sign(S jσ)πn+1S jσ

),

Rg =∑

σ∈5(X1)

n+1∑j=1

sign(S jσ)

n∑k=1

(−1)kπk S jσg for g ∈ Gn+1.

So we have AS ◦ d = Q+ R. We know that

π0S j−1σ = πn+1S jσ for 1≤ j ≤ n,

sign(S j−1σ)π0S j−1σ + (−1)n+1 sign(S jσ)πn+1S jσ

= (−1)n( j−1) sign(σ )π0S j−1σ + (−1)n+1(−1)nj sign(σ )πn+1S jσ = 0.

Thus we get Q = 0.We need the notation σk,k+1 for the flip of k and k+ 1:

σk,k+1 = (1, 2, . . . , k− 2, k− 1, k+ 1, k, k+ 2, k+ 3, . . . , n+ 1) ∈5(X).

Then we get

sign(σk,k+1ρ)πkσk,k+1ρg+ sign(ρ)πkρg = 0 for ρ ∈5(X) and 1≤ k ≤ n.

Hence we come to

R =∑

σ∈5(X1)

n+1∑j=1

sign(S jσ)

n∑k=1

(−1)kπk S jσ

=

n∑k=1

(−1)kn+1∑j=1

∑σ∈5(X1)

sign(S jσ)πk S jσ =

n∑k=1

(−1)k∑

ρ∈5(X)

sign(ρ)πkρ

=

n∑k=1

(−1)k∑

ρ∈50(X)

(sign(ρ)πkρ+ sign(σk,k+1ρ)πkσk,k+1ρ)= 0,

where50(X) is the group of even permutations of X , that is, the alternating group.Therefore we conclude AS ◦ d = Q+ R = 0. ♥

Let A be a G-module with action α. We recall the dimension shifting theoremand the dimension shift map ∂ . First, a new G-module A is defined through thefollowing:


(i) Map(G,A) is the module AG of all A-valued functions on G with pointwiseaddition.

(ii) The group A is the submodule of Map(G,A) of constant A-valued functions.

(iii) The action α of G on A extends to the enlarged additive group Map(G,A) by

(αh f )(g)= αh( f (gh)) for f ∈Map(G,A) and g, h ∈ G.

(iv) Finally A is the quotient G-module A=Map(G,A)/A.

Thus we obtain the equivariant short exact sequence

(2-7) 0→A // Map(G,A) // A→ 0.

The short exact sequence (2-7) splits as follows:

(i) First, set j ( f )(g)= f (g)− f (e) for f ∈Map(G,A) and g∈G, where e∈G isthe neutral element of G. Then the map j is a homomorphism of Map(G,A)onto the subgroup Map0(G,A) of all A-valued functions on G vanishing at e.Then we get Ker( j) = A ⊂ Map(G,A), so that the map j is viewed as abijection from A onto Map0(G,A).

(ii) The map j transforms the action α of G on A to the action, denoted by α again,on Map0(G,A) defined by (αh f )(g) = αh( f (gh))− αh( f (h)) for g, h ∈ Gand f ∈Map0(G,A).

With the map j , we will identify A and Map0(G,A). Thus we have a short exactsequence

0→Ai // Map(G,A)

j←−

s

// A=Map0(G,A)→ 0.

Let s denote the embedding of A=Map0(G,A) ↪→Map(G,A), which is a rightinverse of the map j . If u ∈ Zn−1

α (G, A), then

0= ∂G u = j (∂Gs(u)),

where ∂G means the coboundary operator in Cnα(G,Map(G,A)), so that we have

∂Gs(u)∈Znα(G,A). We denote the cohomology class [∂Gs(u)] ∈Hn

α(G,A) by ∂[u]for each [u] ∈ Hn−1

α (G, A). It is known as the dimension shift theorem that themap ∂ is an isomorphism of Hn−1

α (G, A) onto Hnα(G,A).

Definition 2.3. Suppose that the group G admits a torsion-free central elementz0 ∈ G. A cocycle c ∈ Zn

α(G,A) is said to be of the standard form (relative to thecentral element z0) if

(i) for each k1, . . . , kn ∈ Z and g1, g2, . . . , gn ∈ G,

(2-8) c(zk10 g1, . . . , zkn

0 gn)= αg1(dc(k1; g2, . . . , gn))+ c(g1, g2, . . . , gn);


(ii) the map k ∈ Z 7→ dc(k; g2, g3, . . . , gn) ∈ A belongs to Z1αz0(Z,A) for each

g2, g3, . . . , gn ∈ G, that is,

(2-9) dc(k+ `; g2, g3, . . . , gn)= dc(k; g2, g3, . . . , gn)

+αkz0(dc(`; g2, g3, . . . , gn)).

(iii) For each k ∈ Z and g1, g2, . . . , gn ∈ G, we have

(2-10) (∂Gdc)(k; g1, g2, . . . , gn)= αkz0(c(g1, g2, . . . , gn))− c(g1, g2, . . . , gn).

Remark 2.4. If we choose dc so that

c(z0g1, zk20 g2, . . . , zkn

0 gn)= αg1(dc(g2, g3, . . . , gn))+ c(g1, g2, . . . , gn),

(∂Gdc)(g1, g2, . . . , gn)= αz0(c(g1, g2, . . . , gn))− c(g1, g2, . . . , gn)

and we define dc(k; g2, g3, . . . , gn) inductively by

(2-11) dc(k; g2, g3, . . . , gn)=dc(g2, g3, . . . , gn)+αz0(dc(k−1; g2, g3, . . . , gn)),

Then the cocycle identity (2-8) for c(g2, g3, . . . , gn) and dc(k; g2, g3, . . . , gn) canbe fulfilled automatically.

In the sequel, we often write dc(g2, g3, . . . , gn) for the d-part of a standardcocycle c without referring to the first variable k in dc(k; g2, g3, . . . , gn).

Lemma 2.5. In the above context, every cocycle c ∈ Znα(G,A) is cohomologous to

a cocycle cs of the standard form.

Proof. For n = 1, the cocycle identity c(zk0g) = αg(c(zk

0))+ c(g) for k ∈ Z andg ∈G shows that with dc(k)= c(zk

0) the cochains dc and c satisfy Definition 2.3(i).Now we have

αkz0(c(g))− c(g)= c(zk

0g)− c(zk0)− c(g)= c(g)+αg(c(zk

0))− c(zk0)− c(g)

= αg(dc(k))− dc(k)= (∂Gdc)(k; g),

which shows the property of Definition 2.3(iii) for c and dc.Now assume our claim is valid for 1, . . . , n− 1 and for any G-module {A, α}.Choose an equivariant short exact sequence

0→Ai // M

j←−

s

// A→ 0

such that Hnα(G,M)= {0} for n ≥ 1, and the cross-section s : A→ M is a homo-

morphism of A into M , but is not equivariant. Then ∂Gs :Zn−1α (G, A)→Zn

α(G,A)gives rise to an isomorphism ∂ :Hn−1

α (G, A)→Hnα(G,A). For a standard cocycle


c ∈ Zn−1α (G, A), we set, for each zk1

0 g1, . . . , zkn−10 gn−1 ∈ G,

c(zk10 g1, . . . , zkn−1

0 gn−1)= αg1(s(dc(k1; g2, g3, . . . , gn−1)))

+ s(c(g1, g2, g3, . . . , gn−1)).

Since j (c)= c, we have c = ∂G c ∈ Znα(G,A). We then compute

c(zk10 g1, . . . , z

kn0 gn)= (∂G c)(zk1

0 g1, . . . , zkn0 gn)

= αz

k10 g1

(c(zk2

0 g2, zk30 g3, . . . , z

kn0 gn)

)+

n−1∑j=1

(−1) j c(zk10 g1, . . . , z

k j0 g j z

k j+10 g j+1, . . . ,gn)

+ (−1)n c(zk10 g1, . . . , z

kn−10 gn−1)

= αz

k10 g1

(αg2(s(dc(k2;g3, . . . ,gn)))+ c(g2,g3, . . . ,gn)

)−(αg1g2(s(dc(k1+ k2;g3, . . . ,gn)))+ c(g1g2,g3, . . . ,gn)

)+

n−1∑j=2

(−1) j(αg1(s(dc(k1;g2, . . . ,g j g j+1, . . . ,gn)))

+ c(g1, . . . ,g j g j+1, . . . ,gn))

+ (−1)nαg1

(s(dc(k1;g2,g3, . . . ,gn−1))

)+ (−1)n c(g1,g2,g3, . . . ,gn−1)

= (∂G c)(g1,g2, . . . ,gn)+αzk10 g1(αg2(s(dc(k2;g3, . . . ,gn))))

+αg1

(αk1

z0(c(g2,g3, . . . ,gn))− c(g2,g3, . . . ,gn)

)−αg1g2

(s(dc(k1;g3, . . . ,gn))+α

k1z0(s(dc(k2;g3, . . . ,gn)))

)+

n−1∑j=2

(−1) jαg1(s(dc(k1;g2, . . . ,g j g j+1, . . . ,gn)))

+ (−1)n(αg1(s(dc(k1;g2,g3, . . . ,gn−1)))

)= (∂G c)(g1,g2, . . . ,gn)

+αg1

(αk1

z0(c(g2,g3, . . . ,gn))− c(g2,g3, . . . ,gn)−αg2

(s(dc(k1;g3, . . . ,gn))

)+

n−1∑j=2

(−1) js(dc(k1;g2, . . . ,g j g j+1, . . . ,gn))

+ (−1)n(s(dc(k1;g2,g3, . . . ,gn−1))))

= (∂G c)(g1,g2, . . . ,gn)

+αg1

(αk1

z0

(c(g2,g3, . . . ,gn)− c(g2,g3, . . . ,gn)

)− ∂G(s ◦ dc)(k1;g2,g3, . . . ,gn)

).


Consequently, we get

c(zk10 g1, . . . , zkn

0 gn)= αg1(dc(k1; g2, g3, . . . , gn))+ c(g1, g2, . . . , gn)

with

c(g1, g2, . . . , gn)= (∂G c)(g1, g2, . . . , gn),

dc(g2, g3, . . . , gn)= αz0(c(g2, g3, . . . , gn))− c(g2, g3, . . . , gn)

− ∂G(s ◦ dc)(g2, g3, . . . , gn).

We now check the requirement (2-10) for dc and c:

αz0(c(g1, g2, . . . , gn))− c(g1, g2, . . . , gn)

= αz0(∂G c(g1, g2, . . . , gn))− ∂G c(g1, g2, . . . , gn)

= ∂G

(αz0(c(g1, g2, . . . , gn))− c(g1, g2, . . . , gn)

)= ∂G

(dc(g2, g3, . . . , gn)+ ∂Gs ◦ dc(g2, g3, . . . , gn)

)= ∂Gdc(g2, g3, . . . , gn).

Thus the cocycle c is standard. ♥

We now state the main result on the asymmetrization, which extends the workof Olesen, Pedersen, and Takesaki [Olesen et al. 1980]:

Theorem 2.6. Let Q be a countable torsion-free abelian group.

(i) The asymmetrization AS maps the group Zn(Q,T) of T-valued n-th cocyclesonto the compact group Xn(Q,T) of all asymmetric multicharacters on nvariables of Q.

(ii) The following sequence is exact for each n ∈ N:

1→ Bn(Q,T) // Zn(Q,T)AS // Xn(Q,T)→ 1.

Consequently,

Hn(Zm,T)∼= Xn(Zm,T)∼=

{Tm!/(n!(m−n)!) if m ≥ n0 if m < n.

More generally, if Q is a countable torsion-free abelian group, then the co-homology group Hn(Q,T) is naturally isomorphic to the Pontrjagin–Kampendual of the n-th exterior power Q ∧ Q ∧ · · · ∧ Q of Q.

(iii) The group Xn(Q,T) is a subgroup of Zn(Q,T) such that

Zn(Q,T)= Xn(Q,T)Bn(Q,T), Xn(Q,T)∩Bn(Q,T)= Ker(Power n!),

and AS c = cn! for c ∈ Xn(Q,T).


Remark 2.7. If the group Q has torsion, then the theorem fails as seen in the casethat Q = Zp = Z/pZ for p ≥ 2, H3(Q,T)∼= Zp and X3(Q,T)= {0}.

For the proof, we need some preparation. First, if n = 1, then the claim istrivially true for any abelian group Q with no assumption on torsion. We thenassume the claim is true for cocycle dimension 1, . . . , n− 1 with n ∈ N fixed andfor any torsion-free abelian group Q. With this induction hypothesis, we preparea couple of lemmas for cocycle dimension n.

Lemma 2.8. (i) If M is an abelian group such that a cocycle c ∈ Zn(M,T) isa coboundary if and only if AS c = 1, then the same is true for the productgroup Q = M ×Z.

(ii) If M is an abelian group such that the asymmetrization AS c of each cocyclec ∈ Zn(M,T) is a multicharacter, then the same is true for the product groupQ = M ×Z.

Proof. Let z0 denote the element of Q corresponding to the product decompositionQ = M ×Z, so that every element q ∈ Q is written uniquely in the form q = mzk

0for m ∈ M and k ∈ Z.

(i) In Lemma 2.2, we proved the triviality of the asymmetrization of a cobound-ary. Thus we prove the converse. Suppose AS c = 1 for c ∈ Zn(Q,T). ByLemma 2.5 the cocycle c is cohomologous to a cocycle cs of standard form, andAS cs = AS c = 1 by Lemma 2.2. So we may and do assume that c is standard:

c( p1, p2, . . . , pn)= dc(p2, p3, . . . , pn)`1cM(p1, p2, . . . , pn),

where pi = pi z`i0 ∈ Q = M×Z. As Q does not act on T, the d-part dc is a cocycle

in Zn−1(Q,T).We look at the asymmetrization of c:

(AS c)( p1, p2, . . . , pn)=∏σ∈Sn

(dc(pσ(2), pσ(3), . . . , pσ(n))`σ(1)

× cM(pσ(1), pσ(2), . . . , pσ(n)))sign σ

=

∏σ∈Sn

dc(pσ(2), pσ(3), . . . , pσ(n))`σ(1) sign σ

×

∏σ∈Sn

cM(pσ(1), pσ(2), . . . , pσ(n))sign σ ,

that is,

(2-12) (AS c)( p1, p2, . . . , pn)=∏σ∈Sn

dc(pσ(2), pσ(3), . . . , pσ(n))`σ(1) sign σ

× (AS cM)(p1, p2, . . . , pn).


To compute the first term of the above expression, we take a closer look at thepermutation group Sn . In particular, we have to pay attention to the fact that thefirst term in the variables of dc is missing. To this end, we fix k with 1 ≤ k ≤ n,which represents the missing term in dc, and consider the cyclic permutation

Sn−1(k)= (1, 2, . . . , k− 1, k+ 1, . . . , n) ∈5({1, 2, . . . , k− 1, k+ 1, . . . , n}).

For σ = (k, σ (2), σ (3), . . . , σ (n)) ∈ Sn , define ρ, ρ and σ through

ρ = S(n−k+1)σ

=

(1 2 · · · k− 1 k k+ 1 · · · n

σ(n− k+ 2) σ (n− k+ 3) · · · σ(n) k σ(2) · · · σ(n− k+ 1)

),

ρ =

(1 2 · · · k− 1 k+ 1 · · · n

σ(n− k+ 2) σ (n− k+ 3) · · · σ(n) σ (2) · · · σ(n− k+ 1)

),

σ = Sn−1(k)k−1ρ

=

(1 2 · · · k− 1 k+ 1 · · · n

σ(2) σ (3) · · · σ(k) σ (k+ 1) · · · σ(n)

)= (σ (2), σ (3), . . . , σ (n)).

Then observing sign ρ = sign ρ, we compute

sign σ = sign Sk−1 sign ρ = (−1)(n−1)(k−1) sign ρ

= (−1)(n−1)(k−1) sign(Sn−1(k)n−k) sign σ

= (−1)(n−1)(k−1)+(n−2)(n−k) sign σ = (−1)k−1 sign σ .

Hence the first term of (2-12) becomes∏σ∈Sn

(dc(pσ(2), pσ(3), . . . , pσ(n))

)`σ(1) sign σ

=

n∏k=1

( ∏σ∈Sn−1(k)

(dc(pσ (1), pσ (2), . . . , pσ (n−1))

)sign σ)`k(−1)k−1

=

n∏k=1

((AS dc)(p1, p2, . . . ,

`pk, . . . , pn))`k(−1)k−1

where the notation ` stands for removing the corresponding variable. Thus (2-12)is replaced by

(2-12′) (AS c)( p1, p2, . . . , pn)

=

n∏k=1

((AS dc)(p1, p2, . . . ,

`pk, . . . , pn))`k(−1)k−1

× (AS cM)(p1, p2, . . . , pn).


The condition AS c = 1 yields that AS cM = 1 with `k = 0 for k = 1, . . . , n andAS dc = 1 with `1 = 1 and `k = 0 for k = 2, . . . , n and p1 = e. Hence cM and dc

are both coboundaries by the induction hypothesis. Choose b ∈ Cn−1(M,T) anda ∈ Cn−2(M,T) such that cM = ∂Mb and dc = ∂Ma. Then the cocycle c has theform

c( p1, p2, . . . , pn)= dc(p2, p3, . . . , pn)`1c(p1, p2, . . . , pn)

= ((∂Ma)(p2, p3, . . . , pn))`1 (∂Mb)(p1, p2, . . . , pn).

Setting f ( p1, p2, . . . , pn−1)= a(p2, p3, . . . , pn−1)−`1b(p1, p2, . . . , pn−1) where

pi = z`i0 pi ∈ Q for i = 1, . . . , n− 1, we compute

(∂Q f )( p1, p2, . . . , pn)

= f ( p2, p3, . . . , pn)×

n−1∏k=1

f ( p1, . . . , pk pk+1, . . . , pn)(−1)k

× f ( p1, p2, . . . , pn−1)(−1)n

= a(p3, . . . , pn)−`2a(p3, . . . , pn)

`1+`2

×

n−1∏k=2

a(p2, . . . , pk pk+1, . . . , pn)−`1(−1)k

× a(p2, p3, . . . , pn−1)−`1(−1)n

× b(p2, p3, . . . , pn)×

n−1∏k=1

b(p1, . . . , pk pk+1, . . . , pn)(−1)k

× b(p1, p3, . . . , pn)(−1)n

= a(p3, . . . , pn)`1

n−1∏k=2

a(p2, . . . , pk pk+1, . . . , pn)−`1(−1)k

× a(p2, p3, . . . , pn−1)−`1(−1)n

× (∂Mb)(p1, p2, . . . , pn)

= ((∂Ma)(p2, p3, . . . , pn))`1 (∂Mb)(p1, p2, . . . , pn)

= c( p1, p2, . . . , pn).

Therefore c is a coboundary. This completes the proof of part (i).(ii) Fix a standard cocycle c ∈ Zn(Q,T) by

c( p1, p2, . . . , pn)= dc(p2, p3, . . . , pn)`1c(p1, p2, . . . , pn)

with dc ∈ Zn−1(M,T) and cM ∈ Zn(M,T). Observing that AS cM and AS dc areboth multicharacters by the assumptions, we compute with (2-12′), for q1 = q1zk1

0 ,


(AS c)( p1q1, p2, . . . , pn)

= (AS dc)(p2, . . . , pn)`1+k1

×

n∏j=2

((AS dc)(p1q1, p2, . . . ,

`p j , . . . , pn))` j (−1) j−1

× (AS cM)(p1q1, p2, . . . , pn)

= (AS dc)(p2, . . . , pn)`1

×

n∏j=2

((AS dc)(p1, p2, . . . ,

`p j , . . . , pn))` j (−1) j−1

× (AS dc)(p2, . . . , pn)k1

×

n∏j=2

((AS dc)(q1, p2, . . . ,

`p j , . . . , pn))` j (−1) j−1

× (AS cM)(p1, p2, . . . , pn)(AS cM)(q1, p2, . . . , pn)

= (AS c)( p1, p2, . . . , pn)(AS c)(q1, p2, . . . , pn).

Thus AS c is indeed multiplicative on the first variable, so that it is an asymmetricmulticharacter of Q = M ×Z. ♥

Lemma 2.9. Suppose that c ∈ Zn(Q,T) has a trivial asymmetrization, that is,AS c = 1. Assume the following:

(a) M is a finitely generated subgroup of Q;

(b) a0 is in Q but not M ;

(c) f ∈ Cn−1(M,T) cobounds the restriction cM of c to M , that is, ∂M f = cM .

Then the cochain f has an extension to the subgroup N = 〈M, a0〉 generated byM and a0 such that ∂N f = cN , where cN is the restriction of c to the subgroup N.

Proof. To apply the structure theory of abelian groups, we use the additive groupoperation in the group Q. From the general theory of abelian groups, it follows thatM and N are both free abelian groups and there exists a free basis {z1, z2, . . . , zm}

of N and nonnegative integers {p1, p2, . . . , pr } ⊂ Z+ for 1 ≤ r ≤ m, such thatN = 〈z1, z2, . . . , zm〉 and M = 〈p1z1, . . . , pr zr 〉. With the assumption for n − 1,every (n− 1)-cocycle µ ∈ Zn−1(M,T) is cohomologous to an asymmetric multi-character µa, that is, there exist ai1,i2,...,ir ∈ R such that

µa(g1, g2, . . . , gn−1)=

exp(

2π i∑

i j∈{1,2,...,r}1≤i1<i2<···<in−1≤n−1

ai1,i2,...,in−1

(ei1,M ∧ ei2,M ∧ · · · ∧ ein−1,M

)(g1, g2, . . . , gn−1)

),


where {ei,M : 1 ≤ i ≤ r} is the coordinate system of M relative to the basis{p1z1, . . . , pr zr }. Setting

νa(g1, g2, . . . , gn−1)=

exp(

2π i∑

i j∈{1,2,...,r}1≤i1<i2<···<in−1≤n−1

ai1,i2,...,in−1

pi1 pi2 · · · pin−1

(ei1 ∧ei2 ∧· · ·∧ein−1

)(g1, g2, . . . , gn−1)

),

where {ei : 1 ≤ i ≤ m} is the coordinate system of N in the basis {z1, . . . , zm},we obtain an extension ν of µa. Choose ξ ∈ Cn−2(M,T) so that µ = (∂Mξ)µa,and extend ξ to a cochain ξ ∈ Cn−2(N ,T). Then the second cocycle (∂Nξ)ν givesan extension of the original (n− 1)-cocycle µ ∈ Zn−1(M,T). Thus we obtain thesurjectivity of the restriction map res : µ ∈ Zn−1(N ,T) 7→ µM ∈ Zn−1(M,T), thatis, the exactness of the sequence

Zn−1(N ,T)res // Zn−1(M,T)→ 1.

By induction on generators, Lemma 2.9 yields that the restriction cN of c to Nis a coboundary. Hence there exists ξ ∈ Cn−1(N ,T) such that cN = ∂Nξ . Then wehave ∂M f = cM = ∂MξM , so µM = ξ

−1M f ∈ Zn−1(M,T). By the first arguments,

we can extend µM to an element ν ∈ Zn−1(N ,T). Set f = νξ ∈ Cn−1(N ,T). Thenewly defined cochain f on N extends the original f ∈Cn−1(M,T) and coboundsthe cocycle cN , that is, ∂N f = (∂Nν)(∂Nξ)= ∂Nξ = cN . ♥

We may now complete the proof of Theorem 2.6 by proceeding from cocycledimension 1, . . . , n− 1 to the cocycle dimension n.

Proof of Theorem 2.6. Suppose that c ∈ Zn(Q,T) and AS c = 1. Let {zk : k ∈N} be a sequence of generators of Q and let Mm = 〈z1, z2, . . . , zm〉 for m ∈ N.

The sequence {Mm} is then increasing and Q =⋃

Mm . The triviality assumptionAS c = 1 and Lemma 2.8(i) yield that the restriction cm of the cocycle c to eachMm is a coboundary, so that there exists fm ∈Cn−1(Mm,T) such that cm = ∂Mm fm .The last lemma however allows us to choose the sequence { fm} so that each fm

is an extension of the previous fm−1. Hence the sequence { fm} gives a cochainf ∈ Cn−1(Q,T) such that f |Mm = fm for m ∈ N, and therefore ∂Q f = c. Thuswe conclude that Ker(AS) ⊂ Bn(Q,T). The inclusion Ker(AS) ⊃ Bn(Q,T) wasproved in Lemma 2.2. Hence Ker(AS)= Bn(Q,T).

Lemma 2.8(ii) for {Mm}m∈N yields that the asymmetrization AS c is a multi-character for any c ∈ Zn(Q,T).

Set ca=AS c for an arbitrary cocycle c ∈Zn(Q,T). Then ca ∈ Xn(Q,T). SinceQ is torsion free, the group Xn(Q,T) is indefinitely divisible. So the n!-fold powermapping ξ ∈ Xn(Q,T) 7→ ξ n!

∈ Xn(Q,T) is surjective. But the asymmetrizationAS on Xn(Q,T) is precisely the n!-fold power. Hence there exists ξ ∈ Xn(Q,T)


such that AS ξ = ξ n!= ca. Now we have AS(ξ−1c) = ξ−n!ca = 1. Therefore

ξ−1c ∈ Bn(Q,T). Consequently, we conclude

Zn(Q,T)= Xn(Q,T)Bn(Q,T),

Xn(Q,T)∩Bn(Q,T)= Xn(Q,T)∩Ker(AS)= {c ∈ Xn(Q,T) : cn!= 1}. ♥

Corollary 2.10. If G is a discrete abelian group, then the asymmetrization of everyn-cocycle c ∈ Zn(G,T) is a multicharacter, that is, AS c ∈ Xn(G,T).

Proof. Let F be a free abelian group large enough so that there exists a surjectivehomomorphism π : F→G. Consider the pull back π∗(c) and its asymmetrization,ASπ∗(c)= π∗(AS c). It follows from Theorem 2.6 that the pull back π∗(AS c) isa multicharacter of F ; consequently the original asymmetrization AS c is a multi-character of G. ♥

3. Universal resolution for a countable discrete abelian group

We discuss a universal resolution group for a countable discrete abelian group.We consider only the case that the abelian group under consideration has infinitelymany generators since the finitely generated case can be covered by the infinitegenerator case. Let G = Z<N be the free abelian group of a finite sequences ofintegers, that is, every element g ∈ G is of the form

g = (g1, g2, . . . , gi , . . . , g`, 0, 0, . . . ) for gi ∈ Z,

with `= `(g) ∈ N, the index of the last nonzero term of g ∈ Z<N. With

(3-1) ai = (0, 0, . . . , 0, 1, 0, 0, . . . ),

where the 1 is in the i-th slot, every element g ∈ Z<N is written uniquely

(3-2) g =∑i∈N

ei (g)ai .

We call {ai : i ∈N} the standard basis of Z<N. We also fix a subgroup N of G thatis generated by a sequence {pi ai : i ∈N} with pi ∈ Z+ and i ∈N. We will use thematrix

P =

p1 0 0 · · ·0 p2 0 · · ·0 0 p3 · · ·...

....... . .

, so that N = PZ<N.


Let M be the additive group of upper triangular matrices with integer coeffi-cients, that is,

M =

m =

0 m12 m13 m14 · · ·

0 0 m23 m24 · · ·

0 0 0. . . · · ·

......

.... . . · · ·

: mi, j ∈ Z

,and set e j,k(m)=m jk for j < k and m ∈M . For i < j , let ai ∧a j be the element ofM such that ek,`(ai ∧a j )= δikδ j`, that is, the matrix with only (i, j)-component 1and all others 0; equivalently ai ∧a j with i < j is the (i, j)-matrix unit of M . LetnM be the M-valued second cocycle of G defined by

(3-3)

e j,k(nM(g; h))= e j (g)ek(h) for g, h ∈ G and 1≤ j < k,

nM(g; h)=

0 e1(g)e2(h) e1(g)e3(h) e1(g)e4(h) · · ·0 0 e2(g)e3(h) e2(g)e4(h) · · ·0 0

. . . e3(g)e4(h) · · ·......

. . .... · · ·

.Let H be the group extension of G associated with nM ∈ Z2(G,M):

H = M ×nM G and L = M ×nM N .

The group operation in H is given by (m, g)(n, h)= (m+n+nM(g; h), g+h) for(m, g), (n, h) ∈ H . The inverse (m, g)−1 is given by

(m, g)−1= (−m+ nM(g,−g),−g)

because (0, 0) = (m, g)(m′, g′) = (m + m′ + nM(g; g′), g + g′), g′ = −g andm′ = −m + nM(g; g). To determine the commutator subgroup [H, H ], we take(m, g), (n, h) ∈ H and compute

(m, g)(n, h)(m, g)−1(n, h)−1

= (m, g)(n, h)(−m+ nM(g; g),−g)(−n+ nM(h; h);−h)

= (m+ n+ nM(g, h), g+ h)

× (−m− n+ nM(g; g)+ nM(h; h)+ nM(g; h),−g− h)

= (nM(g; h)+ nM(g; g)+ nM(h; h)+ nM(g; h)+ nM(g+ h;−(g+ h)), 0)

= (nM(g; h)− nM(h; g), 0)

=

(∑j<k

(e j (g)ek(h)− e j (h)ek(g))(a j ∧ ak), 0).

Lemma 3.1. The commutator subgroup [H, H ] of H is the center M of H.


Proof. From the computation above, it follows that for each pair j < k

sH (a j )sH (ak)sH (a j )−1sH (ak)

−1= a j ∧ ak,

with sH the cross-section of π0 : (m, g)∈ H 7→ g ∈G given by sH (g)= (0, g)∈ Hfor g ∈ G. Thus [H, H ] contains the generators a j ∧ ak for j < k of M . ♥

Theorem 3.2. The pair {H, π0} is a universal resolution of the third cocycle groupZ3(G,T) of G. If K is a countable discrete abelian group, then for any surjectivehomomorphism π :Z<N

→ K , the composed map πK = π ◦π0 : H→ K makes thepair {H, πK } a universal resolution of the third cocycle group Z3(K ,T).

Proof. Since Z<N is a free abelian group on countably infinite generators, thereexists a surjective homomorphism from G to any countable abelian group K . Soit is sufficient to prove that

π∗0 (Z3(G,T))⊂ B3(H,T).

For each triplet ξ, η, ζ ∈Hom(G,R), we define a multihomomorphism, called thetensor product and denoted by ξ ⊗ η⊗ ζ ∈ C3(G,R), as follows:

(ξ ⊗ η⊗ ζ )(g; h; k)= ξ(g)η(h)ζ(k) for g, h, k ∈ G.

Then the tensor product ξ ⊗ η⊗ ζ generates the third cocycle group Z3(G,R) upto coboundary, that is,⟨{

ξ ⊗ η⊗ ζ : ξ, η, ζ ∈ Hom(G,R)}⟩+B3(G,R)= Z3(G,R).

Now for each pair η, ζ ∈ Hom(G,R), we define a cochain Bη,ζ ∈ C1(H,R) by

(3-4) Bη,ζ (g)=∑j<k

η(a j )ζ(ak)e j,k(m0(g)) for g = (m0(g), π0(g)) ∈ H.

Then we have(∂H(π

∗

0 ξ ⊗ Bη,ζ ))(g1; g2; g3)

= ξ(π0(g2))Bη,ζ (g3)− ξ(π0(g1)+π0(g2))Bη,ζ (g3)

+ ξ(π0(g1))Bη,ζ (g2g3)− ξ(π0(g1))Bη,ζ (g2)

=−ξ(π0(g1))Bη,ζ (g3)+ ξ(π0(g1))(∑

j<k η(a j )ζ(ak)e j,k(m0(g2g3)))

− ξ(π0(g1))Bη,ζ (g2)

=−ξ(π0(g1))Bη,ζ (g3)

+ ξ(π0(g1))(∑

j<k η(a j )ζ(ak)(e j,k(m0(g2)+m0(g3)+π∗

0 nM(g2; g3)))

− ξ(π0(g1)Bη,ζ (g2)

= ξ(π0(g1))(∑

j<k η(a j )ζ(ak)e j (π0(g2))ek(π0(g3))).


Choosing ξ, η, ζ ∈ Hom(G,T) to be ξ = ei , η = e j and ζ = ek for i < j < k, weobtain

π∗0 (ei ⊗ e j ⊗ ek)= ∂H(π∗

0 ei ⊗ Be j ,ek ).

Every third cocycle in Z3(G,T) is cohomologous to a cocycle ca ∈ Z3(G,T) ofthe form

(3-5) ca(g1; g2; g3)= exp(

2π i( ∑

i< j<k

a(i, j, k)ei (g1)e j (g2)ek(g3))).

So with ba ∈ C2(H,T) defined by

(3-6) ba(g1; g2)= exp(

2π i( ∑

i< j<k

a(i, j, k)ei (π0(g1))Be j ,ek (g3))),

we have

(3-7) π∗0 ca = ∂H ba.

Hence we get π∗0 (Z3(G,T)) ⊂ B3(H,T), from which we conclude that the pair

{H, π0} is a universal resolution of Z3(G,T). ♥

Remark 3.3. The µ-part of every characteristic cocycle (λ, µ) ∈ Z(H,M,T) istrivial.

Proof. Since MGH is central, λ is a bicharacter of M×H ; in particular λ(m, · ) is acharacter of H for every m∈M . Hence it must vanish on the commutator subgroup,that is, λ(m, n)= 1 for all m, n ∈ M . Thus µ ∈ Z2(M,T) is a coboundary. ♥

Consider (λ, µ) ∈ Z(H, L ,T) with L = M ×nM N . We may and do assume thetriviality µM = 1 of the restriction of µ to M . We then have the correspondingcrossed extension

1→ T // Ej←−

u

// L→ 1

The triviality of µM means that the cross-section u is multiplicative on M , that is,u(mn)= u(m)u(n) for m, n ∈ M . Here we use the multiplicative group operationsince M sits in the noncommutative group H .

Lemma 3.4. If sH is a cross-section of the quotient map π0 : H → Z<N= H/M

with nM = ∂sH ∈ Z2(Z<N,M), then each characteristic cocycle in Z(H, L ,M,T)

is cohomologous to the one (λ, µ) ∈ Z(H, L ,M,T) such that

λ(m; nsH (h))= λ(m; sH (h)) for m, n ∈ M, h ∈ Z<N,

µ(msH (g); nsH (h))= λ(n; sH (g))µ(sH (g); sH (h)) for m, n ∈ M, g, h ∈ N .


Proof. In the crossed extension E ∈ Xext(Hm, L ,M,T) associated with (λ, µ) ∈Z(Hm, L ,M,T) given by 1→ T→ E→ L→ 1, we redefine the cross-section ufor m ∈ M and g ∈ N as u(msH (g)) = u(m)u(sH (g)), so that µ(m; sH (g)) = 1.We now compute, for m, n ∈ M and h ∈ Z<N,

λ(m; nsH (h))u(m)= αnsH (h)(u(m))= u(n)αsH (h)(u(m))u(n)−1

= λ(m; sH (h))u(mn)u(n)−1

= λ(m; sH (h))u(m);

for g, h ∈ N , we complete the proof with the computation

µ(msH (g); nsH (h))u(msH (g)nsH (h))

= u(msH (g))u(nsH (h))

= u(m)u(sH (g))u(n)u(sH (h)))

= u(m)αsH (g)(u(n))u(sH (g))u(sH (h))

= λ(n; sH (g))u(m)u(n)µ(sH (g); sH (h))u(sH (g)sH (h))

= λ(n; sH (g))u(mn)µ(sH (g); sH (h))u(sH (g)sH (h))

= λ(n; sH (g))µ(sH (g); sH (h))u(msH (g)nsH (h)). ♥

Groups G, Hm, Gm and Qm. First, we fix notations. To work on the quotientgroup Z/pZ= Zp with p ∈ N and p ≥ 2, we set

(3-8)[i]p = i + pZ ∈ Zp, where i = np+{i}p, 0≤ {i}p < p,

ηp([i]p, [ j]p)= {i}p +{ j}p −{i + j}p ={

0 if {i}p +{ j}p < p,p if {i}p +{ j}p ≥ p.

We shall call the pZ-valued cocycle ηp ∈ Z2(Zp, pZ) the Gauss cocycle, whichcan be written

(3-8′) ηp([i]p, [ j]p)= p([ i + j

p

]−

[ ip

]−

[ jp

]),

where [x] for x ∈ R is the largest integer less than or equal to x .Given a homomorphism m of the group G to R/T ′Z such that Ker(m)⊃ N , we

consider the group extension

Gm = {(g, s) ∈ G×R : sT ′ = s+ T ′Z=m(g) ∈ R/T ′Z},

0→ Zn→zn

0 // Gmπm // G→ 1,


where z0 = (0, T ′) ∈ Gm. Identifying m with m ◦ π0 ∈ Hom(H,R/T ′Z), we alsoform a group extension

Hm = {(h, s) ∈ H ×R :m(h)= sT ′ ∈ R/T ′Z}

= {(m, g, s) ∈ M ×G×R :m(g)= sT ′ ∈ R/T ′Z},

0→ Zn→zn

0 // Hm // H → 1,

where the central element z0= (1, T ′)∈ Hm appears in both Gm and Hm. We hopethat this abuse of notation for two distinct elements in the different groups will notcause a headache later; it is just like the zero elements in ring theory.

By the assumption N ⊂Ker(m), the homomorphism m factors through the quo-tient group Q=G/N , so that it is also viewed as a homomorphism of Q→R/T ′Z;therefore we can form the group extension Qm as before, which sits on the follow-ing commutative diagram of exact sequences:

1

��

1

��0

��

// sm(N )

��

// N

��

// 0

0 // Z // Gm

πQ ↑s

��

πm

←−sm

// G

πQ ↑s

��

// 1

0 // Z

��

// Qm

��

πm

←−sm

// Q

��

// 1

0 1 1

From the assumption Ker(m) ⊃ N , it follows that m(pi ai ) = 0, so that thereexists an integer qi ∈ Z with 0≤ qi < pi such that

(3-9)mi = {m(ai )}T ′ = qi T ′/pi ∈ ((T ′/pi )Z),

m(ai )= mi =mi +T′Z ∈ R/T ′Z.

For g ∈ G, we set

(3-10)

Gm 3 zi =

{(ai ,mi ) if i 6= 0,(0, T ′) if i = 0,

sm(g)=∑i∈N

ei (g)zi =

(g,∑i∈N

ei (g)mi

)= (g, n(g)),

n(g)=∑i∈N

ei (g)mi .


Then Gm decomposes as

(3-11)

Gm = Zz0⊕ sm(G)=∑⊕

i∈N0

Zzi , where N0 = N∪ {0},

g = e0(g)z0+∑i∈N

ei (g)zi ∈ Gm;

g = (g, s)= (0, e0(g, s)T ′)+(∑

i∈N

ei (g)ai ,∑i∈N

ei (g)mi

)= (0, e0(g, s)T ′)+

∑i∈N

ei (g)zi ;

e0(g, s)= (s− n(g))/T ′ ∈ Z,

ei (g, s)= ei (g) for i ∈ N.

In particular, if g ∈ N , we have g= (g, 0)=−(n(g)/T ′)z0+∑

i∈N ei (g)zi , so that

e0(g)=−n(g)/T ′ 6= 0 unless n(g)=∑i∈N

ei (g)mi = 0.

We then have m(g) = [n(g)]T ′ ∈ R/T ′Z. Setting b j = p j a j for j ∈ N, we writeevery g ∈ N uniquely in the form

(3-12) g =∑j∈N

e j (g)p j

b j =∑j∈N

e j,N (g)b j ,

where e j,N (g)= e j (g)/p j ; also in Hm we have

(3-13) b j = p j z j − p j m j z0 = p j z j − q j z0.

Remark. The element (ai , 0) is not a member of Gm.

Next we define a cross-section sm : Q→ Qm so that the diagramGm G

smoo

Qm

s

OO

Qsm

oo

s

OO

commutes. First, we set

g = g+ N ∈ Q = G/N for g ∈ G, s(q)=∑i∈N

{ei (q)}pi ai for q ∈ Q,

ai = πQm(ai ), zi = (ai ,mi ),

sm(q)=∑i∈N

{ei (q)}pi zi =∑i∈N

{ei (q)}pi (ai ,mi )=(

q,∑i∈N

{ei (q)}pi mi

),

s(q, s)= (s(q), s) ∈ Gm for (q, s) ∈ Qm.


The cross-section s : Qm→ Gm gives rise to an N -valued cocycle

(3-14) nN = ∂Qs ∈ Z2(Qm, N ),

which is given by

nN(q1; q2)= s(q1, s1)+ s(q2, s2)− s(q1+ q2, s1+ s2)

= (s(q1)+ s(q2)− s(q1+ q2), 0)

=

(∑i∈N

ηpi ([ei (q1)]pi ; [ei (q2)]pi )ai , 0)

=

∑i∈N

(ηpi ([ei (q1)]pi ; [ei (q2)]pi )ai , 0) ∈ N = N ×{0}

for each pair q1 = (q1, s1), q2 = (q2, s2) ∈ Qm.For each element h = (m, g) ∈ H with m ∈ M and g ∈ G, we write m =m0(h)

and g = πG(h). Then we have L = π−1G (N ) and

m0(gh)= m0(g)+m0(h)+ nM(πG(g);πG(h)) for g, h ∈ H.

For short, we write ei, j (g)= ei, j (m0(g)) for g = (m0(g), g, s) ∈ Hm and i, j ∈N.With sH (g)= (0, g) ∈ H for each g ∈ G, we have

nM(g; h)= sH (g)+ sH (h)− sH (g+ h)= ∂GsH (g; h) for g, h ∈ G.

With s = sH ◦ s, we obtain a cross-section s of πQ ◦ πG : H → Q = H/L , whichgives rise to an L-valued second cocycle nL ∈ Z2(Q, L); for q1, q2 ∈ Q, it is

(3-15)

nL(q1; q2)= s(q1)s(q2)s(q1+ q2)−1

= sH(s(q1)

)sH(s(q2)

)sH(s(q1+ q2)

)−1

= nM

(s(q1); s(q2)

)sH(s(q1)+ s(q2)

)sH(s(q1+ q2)

)−1

= nM

(s(q1); s(q2)

)sH(nN(q1; q2)+ s(q1+ q2)

)sH(s(q1+ q2)

)−1

= nM

(s(q1); s(q2)

)nM

(nN(q1; q2); s(q1+ q2)

)−1sH

(nN(q1; q2)

).

We further compute the ( j, k)- and k-components as

(3-16)

e j,k(nM(s(q1); s(q2))

)= e j (s(q1))ek(s(q2))

= {e j (q1)}p j {ek(q2)}pk ,

e j,k(nM(nN(q1; q2); s(q1+ q2))

)= e j (nN(q1; q2))ek(s(q1+ q2))

= ηp j ([e j (q1)]p j ; [e j (q2)]p j ){ek(q1+ q2)}pk ,

ek(sH (nN(q1; q2))

)= ηpk ([ek(q1)]pk ; [ek(q2)]pk ).


SinceHm = M ×π∗m(nM )

(∑⊕

i∈N

Zzi ⊕Zz0

),

for each h = (m, g) ∈ H , we set

(3-17) sm(h)= (m, sm(g))=(

m,∑i∈N

ei (g)zi

)=

(m, g,

∑i∈N

ei (g)mi

),

and we identify ` = (m, Pg) ∈ L with (m, Pg, 0) ∈ Hm, so that L is a subgroupof Hm, while H is not.

4. The characteristic cohomology group 3(Hm, L, M,T)

Since H is a universal resolution group for G=Z<N, every third cohomology class[c] ∈ H3(G,T) is of the form [c] = δHJR[λ,µ] for some [λ,µ] ∈3(H,M,T). Soevery outer action α of G on a factor M of type IIIλ comes from an action α of H ,that is, the outer action α is given by

(4-1) αg = αsH (g) for g ∈ G.

But the action α of H does not give rise to an action of H on the reduced (discrete)core Md . Instead, the action α of H on M gives rise naturally to an action, denotedby the same notation α, of Hm on Md , where

m(h)= mod (αh) ∈ R/T ′Z for h ∈ H.

If N = α−1(Cntr(M)))⊂G, then L =α−1(Cntr(M)). We make a basic assumptionon the subgroup N that

N = PG = PZ<N.

In the case that G is finitely generated free abelian group, the fundamental structuretheorem for finitely generated abelian groups guarantees that every subgroup of Gis of this form.

We study first the characteristic cohomology group 3(Hm, L ,M,T) and mod-ified HJR-map δ :3(Hm, L ,M,T)→ Hout

m,s(G, N ,T).We introduce a series of notations first:

(4-2)

N0 = N∪ {0} = Z+,

10 = {(i, j, k) ∈ N30 : i < j < k} ∪ {(i, i, k) ∈ N3

0; i < k}∪ {(k, i, k) ∈ N3

0 : i < k},1=10 ∩N3.

For each g ∈ Hm, let m0(g) be the M-component of g, that is,

(4-3) m0(g)= gsH (πG(g))−1∈ M for g ∈ Hm.


We regard ei and e j,k as functions defined on Hm by fixing the coordinate system

(4-4) g =( ∑

1≤ j<k

e j,k(g)(a j ∧ ak),∑i∈N0

ei (g)zi

)∈ Hm, with g = πm(g) ∈ H.

We then introduce a cochain B jk ∈ C1(Hm,R) defined for h ∈ Hm by

(4-5) B jk(h)=

−e j,k(m0(h)) if j < k,−

12(e j e j )(h) if j = k,

ek, j (m0(h))− (e j ek)(h) if j > k,

The cochain enjoys the property

(4-6) ∂H B jk = π∗

0 (e j ⊗ ek) for j, k ∈ N.

We continue to define the following cochains for each a ∈ RN30 :

Xa(i, j, k)= a(i, j, k)e j,k ⊗ ei + a( j, i, k)ei,k ⊗ e j + a(k, i, j)ei, j ⊗ ek,

Xa(i, k)= a(i, i, k)ei,k ⊗ ei + a(k, i, k)ei,k ⊗ ek,

Ya(i, j, k)= a(i, j, k)(Bi j ⊗ ek + ek ⊗ B j i − Bik ⊗ e j − e j ⊗ Bki

)+ a( j, i, k)

(B j i ⊗ ek + ek ⊗ Bi j − B jk ⊗ ei − ei ⊗ Bk j

)+ a(k, i, j)

(Bki ⊗ e j + e j ⊗ Bik − Bk j ⊗ ei − ei ⊗ B jk

),

Ya(i, k)= a(i, i, k)(Bi i ⊗ ek + ek ⊗ Bi i − Bik ⊗ ei − ei ⊗ Bki )

+ a(k, i, k)(Bki ⊗ ek + ek ⊗ Bik − Bkk ⊗ ei − ei ⊗ Bkk),

Z( · · · )(g; h)= Y ( · · · )(m0(h); g),

Za(i, j, k)= a(i, j, k)(e j ⊗ ei,k − ek ⊗ ei, j

)+ a( j, i, k)

(ek ⊗ ei, j + ei ⊗ e j,k

)+ a(k, i, j)

(e j ⊗ ei,k − ei ⊗ e j,k

),

Za(i, k)= a(i, i, k)ei ⊗ ei,k + a(k, i, k)ek ⊗ ei,k;

fi, j,k = 2(ei e j )⊗ ek − 3ei ⊗ (e j ek)+ e j ⊗ (ei ek)

− 2(ei ek)⊗ e j − ek ⊗ (ei e j ),

Ua(i, j, k)= 16

(a(i, j, k) fi, j,k + a( j, i, k) f j,i,k + a(k, i, j) fk,i, j

− (AS a)(i, j, k) fi, j,k),

Ua(i, k)=−a(i, i, k)Bi i ⊗ ek + a(k, i, k)(Bkk ⊗ ei − ek ⊗ (ei ek)),

Va(i, j, k)= Za(i, j, k)+π∗GUa(i, j, k),

Va(i, k)= Za(i, k)+π∗GUa(i, k).


The infinite summations

(4-7)

Xa =∑

i< j<k

Xa(i, j, k)+∑i<k

Xa(i, k),

Ya =∑

i< j<k

Ya(i, j, k)+∑i<k

Ya(i, k),

Ua =∑

i< j<k

Ua(i, j, k)+∑i<k

Ua(i, k),

Va =∑

i< j<k

Va(i, j, k)+∑i<k

Va(i, k),

Za =∑

i< j<k

Za(i, j, k)+∑i<k

Za(i, k)

will become all finite sums as soon as variables from M or Hm are fed in. So nodivergence problem in the infinite sums will occur.

The cochain fi, j,k relates basic cocycles ei⊗e j⊗ek and the asymmetric trichar-acter

deti jk = (ei ⊗ e j ⊗ ek + e j ⊗ ek ⊗ ei + ek ⊗ ei ⊗ e j )

− (e j ⊗ ei ⊗ ek + ei ⊗ ek ⊗ e j + ek ⊗ e j ⊗ ei )= ei ∧ e j ∧ ek

as

(4-8) deti jk = ∂L fi, j,k + 6ei ⊗ e j ⊗ ek for i < j < k,

which can be confirmed by a direct computation.Let Z be the set of all pairs (a, b) of functions a : (i, j, k)∈N3

7→ a(i, j, k)∈R

and b : (i, j) ∈ N20 7→ b(i, j) ∈ R such that a satisfies

(4-9Z-a)

a(i, j, k)= 0 for j, k ∈ N0 with j ≥ k,

a(0, j, k)= 0 for every j, k ∈ N0,

(AS a)(i, j, k)= a(i, j, k)− a( j, i, k)+ a(k, i, j)

∈

( 1gcd(pi , p j , pk)

Z).

and b satisfies

(4-9Z-b)b(i, j)p j − b(i, 0)q j ∈ Z for i, j ∈ N with i < j,

b(0, j)= 0 for j ∈ N0.

Let Za be the set of a ∈RN3satisfying (4-9Z-a), and let Zb be the set of all b ∈RN2

0

satisfying (4-9Z-b). So we have Z= Za ⊕Zb.


Let B be the subgroup of Z consisting of all those (a, b)∈Z such that a satisfiesthe coboundary condition

(4-9B-a)

a(i, j, k), a(k, i, j), a( j, i, k) ∈ Z if i < j < k,

a(i, i, k) ∈ 2Z if i < k,

a(k, i, k) ∈ 2Z if i < k,

and b satisfies the coboundary condition

(4-9B-b)

b(i, j)pi+

b( j, i)p j∈

( 1pi

Z)+

( 1p j

Z)=

( 1lcm(pi , p j )

Z)

if i < j,

b(i, 0) ∈ Z and b(i, i) ∈ Z if i ∈ N.

Let Ba (respectively Bb) be the set of all b ∈ RN20 satisfying (4-9B-a) (respectively

(4-9B-b)). Thus we have B = Ba ⊕Bb. Set Ha = Za/Ba and Hb = Zb/Bb. WithD(i, j, k)= gcd(pi , p j , pk) for each triplet i < j < k with i, j, k ∈ N, we set

Za(i, j, k)= {(u, v, w) ∈ R3: u− v+w ∈ ((1/D(i, j, k))Z)},

Ba(i, j, k)= Z⊕Z⊕Z,

where u = a(i, j, k), v = a( j, i, k) and w = a(k, i, j). For a pair i, k ∈ N withi < k, we set

Za(i, k)= {(x, y) ∈ R2} = R⊕R and Ba(i, k)= (2Z)⊕ (2Z),

where x = a(i, i, k) and y = a(k, i, k). We then naturally define

3a(i, j, k)= Za(i, j, k)/Ba(i, j, k)

∼=

(( 1D(i, j, k)

Z) /

Z)⊕R/Z⊕R/Z for i < j < k,

3a(i, k)= Za(i, k)/Ba(i, k)= R/(2Z)⊕R/(2Z) for i < k.

Here the second isomorphism above can be seen easily by considering the matrix

A =

1 −1 10 1 00 0 1

∈ SL(3,Z).

For each ordered pair i, j ∈ N with i < j , we put Di, j = gcd(pi , p j ) and define

(4-10)

Zb(i, j)= {(x, u, y, v) ∈ R4: p j x − q j u ∈ Z, pi y− qiv ∈ Z},

Bb(i, j)= {(x, u, y, v) ∈ Zb(i, j) : p j x + pi y ∈ Di, j Z, u, v ∈ Z},

Zb(i, i)= {z = (x, u) ∈ R2: pi x − qi u ∈ Z}, Bb(i, i)= Z⊕Z,

3b(i, j)= Zb(i, j)/Bb(i, j), 3b(i, i)= Zb(i, i)/Bb(i, i).


Definition 4.1. To each (a, b) ∈ Z we associate a cochain (λa,b, µa) defined by

(4-11)

λa,b(g; h)= exp(2π i((Ya + XAS a)(g; h)))

× exp(2π i

(∑i∈N, j∈N0

b(i, j)ei,N (g)e j (h))),

ηa(g; h)= exp(2π i(Ya(g; h))),

µa(g; h)= exp(2π iVa(g; h))

= λa,b(m0(h); g) exp(2π iUa(πG(g);πG(h)))

for each (g, h)∈ L×Hm. In the case that b= 0 (respectively a = 0) we denote thecorresponding cochains by (λa, µa) (respectively λb). Let Za (respectively Ba) bethe set of {(λa, µa) : a ∈Za} (respectively {(λb, 1) : b ∈Zb}), and let3=3a⊕3b,3a = Za/Ba and 3b = Zb/Bb.

Theorem 4.2. (a) The cochain (λa, µa) is a characteristic cocycle belonging toZ(Hm, L ,M,T) and the correspondence a ∈ Za 7→ (λa, µa) ∈ Za gives thefollowing commutative diagram of exact sequences:

(4-12a)

0

��0→ Ba

��

// a ∈ Za

��

// [a] ∈ Ha→ 0

��0→ Ba // (λa, µa) ∈ Za // [λa, µa] ∈3a→ 1

��0

(b) The correspondence b ∈ Zb 7→ (λb, 1) ∈ Zb gives the following commutativediagram of exact sequences:

(4-12b)

0

��0→ Bb

��

// b ∈ Zb

��

// [b] ∈ Hb→ 0

��0→ Bb // (λb, 1) ∈ Zb // [λb] ∈3b→ 1

��0

(c) The characteristic cohomology group3(Hm, L ,M,T)=3a⊕3b has furtherfine structure:


(i) The group 3a has the Cartesian product decomposition

(4-13a) 3a =∏

i< j<k

3a(i, j, k)⊕∏i< j

3a(i, j),

where 3a(i, j, k)∼= ZD(i, j,k)⊕R/Z⊕R/Z,

D(i, j, k)= gcd(pi , p j , pk),

3a(i, j)∼= R/(2Z)⊕R/(2Z).

(ii) The fiber product decomposition of 3b is the family {3b(i, j) : i, j ∈ N}

and each group 3b(i, j) is described by

(4-13a)3b(i, j)∼= Z/(gcd(pi , p j , qi , q j )Z)⊕ (R/Z)⊕ (R/Z) for i < j,

3b(i, i)∼= Z/(gcd(pi , qi )Z)⊕ (R/Z).

The group 3b(i, j) is equipped with three homomorphisms, and 3b(i, i)has two:

(4-14)

πi j :3b(i, j)→( 1

D(i, j)Z)/

Z,

π ii, j :3b(i, j)→ R/Z, π

ji, j :3b(i, j)→ R/Z,

πi i :3b(i, i)→ (1/pi )Z/Z, π ii :3b(i, i)→ R/Z,

These are such that for each z = (x, u, y, v) ∈ Zb(i, j)

(4-15)

πi j ([λz])= [mi, j (xr j,i + yri, j )− ni, j (us j,i + vsi, j )]Z,

π ii, j ([λz])= [u]Z, π

ji, j ([λz])= [v]Z,

πi i ([λz])= [pi x − qi u]Z, π ii ([λz])= [u]Z,

where

(4-16)

D(i, j)= gcd(pi , p j , qi , q j ),

Di, j = gcd(pi , p j ), Ei, j = gcd(qi , q j ),

ri, j = pi/Di, j , r j,i = p j/Di, j si, j = qi/Ei, j , s j,i = q j/Ei, j ,

mi, j = Di, j/D(i, j), ni, j = Ei, j/D(i, j),

qiwi, j + q jw j,i = Ei, j , xi, j Di, j + yi, j Ei, j = D(i, j).

The group 3b is the fiber product of {3b(i, j) : i, j ∈ N} relative to themaps {π i

i, j , πj

i, j , πii : i, j ∈N} in the sense that3b is the group of all those

λb ∈∏(i, j)∈N2 3b(i, j) such that

(4-17) π ii, j [λb(i, j)] = π i

i [λb(i, i)] = π iki [λb(k, i)] for i, j, k ∈ N.


We will prove the theorem in several steps.First, we observe that the asymmetrization of fi, j,k is given by

(4-18)

AS fi, j,k = 2(ei e j )∧ ek − 3ei ∧ (e j ek)+ e j ∧ (ei ek)

− 2(ei ek)∧ e j − ek ∧ (ei e j )

= 3((e j ek)∧ ei − (ei ek)∧ e j + (ei e j )∧ ek

).

Lemma 4.3. (i) The difference Xa−Ya is equal to XAS a on M × Hm. In partic-ular, if the integers

ei, j (m)ek(g), e j,k(m)ei (g), ei,k(m)e j (g), e j,k(m)ei (g)

are all divisible by gcd(pi , p j , pk), then for each a ∈ Z

Ya(i, j, k)(m; g)≡ Xa(i, j, k)(m; g) mod Z for m ∈ M and g ∈ Hm.

Therefore, if either g ∈ L or m ∈ L ∧ Hm, then

(4-19)Xa(i, j, k)(m; g)≡ Ya(i, j, k)(m; g) mod Z,

Xa(i, j, k)(h1 ∧ g; h2)≡ Ya(i, j, k)(h1 ∧ g; h2) mod Z

for each h1, h2 ∈ Hm.

(ii) For every m ∈ M and g ∈ Hm and i < k we have

(4-20) Xa(i, k)(m; g)= Ya(i, k)(m; g).

Proof. (i) We simply compute for i < j < k:

(Xa(i, j, k)− Ya(i, j, k))(m; g)

= a(i, j, k)e j,k(m)ei (g)+ a( j, i, k)ei,k(m)e j (g)

+ a(k, i, j)ei, j (m)ek(g)

− a(i, j, k)(ei,k(m)e j (g)− ei, j (m)ek(g))

− a( j, i, k)(ei, j (m)ek(g)+ e j,k(m)ei (g))

− a(k, i, j)(ei,k(m)e j (g)− e j,k(m)ei (g))

= (a(i, j, k)− a( j, i, k)+ a(k, i, j))

× (e j,k(m)ei (g)− ei,k(m)e j (g)+ ei, j (m)ek(g)).

Thus we conclude (Xa − Ya)(m; g)= XAS a(m; g) for m ∈ M and g ∈ Hm.

(ii) The assertion follows from an easy direct computation. ♥


Lemma 4.4. If a ∈ R1 is asymmetric modulo ((1/(pi p j pk))Z) in that

(4-21) (AS a)(i, j, k)= a(i, j, k)− a( j, i, k)+ a(k, i, j) ∈ ((1/(pi p j pk))Z)

for each triplet i < j < k, then the cochain µa of (4-11), that is,

µa(g; h)= exp(2π i(Va(g; h))) for g, h ∈ L ,

is a second cocycle µa ∈ Z2(L ,T).

Proof. Observing

(∂Lµa)(g1; g2; g3)= exp(2π i(∂L Va(g1; g2; g3))) for g1, g2, g3 ∈ L ,

we compute the coboundary of Va:

∂L Va(i, j, k)= ∂L Za(i, j, k)+ ∂LUa(i, j, k)

= a(i, j, k)(e j ⊗ ei ⊗ ek − ek ⊗ ei ⊗ e j )

+ a( j, i, k)(ek ⊗ ei ⊗ e j + ei ⊗ e j ⊗ ek)

+ a(k, i, j)(e j ⊗ ei ⊗ ek − ei ⊗ e j ⊗ ek)

+16∂L

(a(i, j, k) fi, j,k + a( j, i, k) f j,i,k + a(k, i, j) fk,i, j

− (AS a)(i, j, k) fi, j,k)

= a(i, j, k)(e j ⊗ ei ⊗ ek − ek ⊗ ei ⊗ e j )

+ a( j, i, k)(ek ⊗ ei ⊗ e j + ei ⊗ e j ⊗ ek)

+ a(k, i, j)(e j ⊗ ei ⊗ ek − ei ⊗ e j ⊗ ek)

+16

(a(i, j, k)(deti jk −6ei ⊗ e j ⊗ ek)

+ a( j, i, k)(det j ik −6e j ⊗ ei ⊗ ek)

+ a(k, i, j)(detki j −6ek ⊗ ei ⊗ e j )

− (AS a)(i, j, k)(deti jk −6ei ⊗ e j ⊗ ek))

≡−(AS a)(i, j, k)(ei ⊗ e j ⊗ ek − e j ⊗ ei ⊗ ek + ek ⊗ ei ⊗ e j )

≡ 0 mod Z on L × L × L ,

since ei ⊗ e j ⊗ ek takes values in pi p j pkZ on L × L × L . Also we have

∂L Va(i, k)= ∂L Za(i, k)+ ∂LUa(i, k)

= a(i, i, k)ei ⊗ ei ⊗ ek + a(k, i, k)ek ⊗ ei ⊗ ek − a(i, i, k)ei ⊗ ei ⊗ ek

+ a(k, i, k)(ek ⊗ ek ⊗ ei − ek ⊗ (ei ⊗ ek + ek ⊗ ei ))

= 0.

Hence µa is a second cocycle on L . ♥


Lemma 4.5. (i) For every (a, b) ∈ Z, the pair {λa,b, µa} is a characteristic co-cycle in Z(Hm, L ,M,T).

(ii) Every characteristic cocycle (λ, µ) ∈ Z(Hm, L ,M,T) is cohomologous tosome (λa,b, µa).

(iii) The characteristic cocycle {λa,b, µa} ∈ Z(Hm, L ,M,T) is a coboundary ifand only if (a, b) ∈ B.

Proof. (i) We first check the cocycle identities for g, g1, g2∈ L and h, h1, h2∈Hm:

((∂L ⊗ id)λa,b)(g1; g2; h)= µa(h−1g1h; h−1g2h)/µa(g1; g2)(a)

= λa,b(g2 ∧ h; g1),

((id⊗∂Hm)λa,b)(g; h1; h2)= 1/λa,b(g∧ h1; h2)(b)

= λa,b(h1 ∧ g; h2),

λa,b(g; h)= µa(h; h−1gh)/µa(g; h) for g, h ∈ L .(c)

Second, we compute for g1, g2 ∈ L and h ∈ Hm that

Xa(i, j, k)(g2 ∧ h; g1)

= a(i, j, k)e j,k(g2 ∧ h)ei (g1)+ a( j, i, k)ei,k(g2 ∧ h)e j (g1)

+ a(k, i, j)ei, j (g2 ∧ h)ek(g1)

= a(i, j, k)ei (g1)(e j (g2)ek(h)− ek(g2)e j (h))

+ a( j, i, k)e j (g1)(ei (g2)ei (h)− ek(g2)ei (h))

+ a(k, i, j)ek(g1)(ei (g2)e j (h)− e j (g2)ei (h))

=(a(i, j, k)ei ⊗ (e j ⊗ ek − ek ⊗ e j )+ a( j, i, k)e j ⊗ (ei ⊗ ei − ek ⊗ ei )

+ a(k, i, j)ek ⊗(ei ⊗ e j − e j ⊗ ei

))(g1; g2; h).

On the other hand, we have

(4-22) (∂L ⊗ id)Ya(i, j, k)= a(i, j, k)(ei ⊗ e j ⊗ ek − ei ⊗ ek ⊗ e j )

+ a( j, i, k)(e j ⊗ ei ⊗ ek − e j ⊗ ek ⊗ ei )

+ a(k, i, j)(ek ⊗ ei ⊗ e j − ek ⊗ e j ⊗ ei ).

Since XAS a(i, j, k)(g2∧h; g1)≡ 0 mod Z, Lemma 4.3 yields, for each g1, g2 ∈ Land h ∈ Hm,(

(∂L ⊗ id)Ya(i, j, k))(g1; g2; h)= Xa(i, j, k)(g2 ∧ h; g1)

≡ Ya(i, j, k)(g2 ∧ h; g1) mod Z.

Similarly, we have

((∂L ⊗ id)Ya(i, k))(g1, g2; h)≡ Ya(i, k)(g2 ∧ h; g1) mod Z.


Next, we have

(4-23)

Xa(i, j, k)(h1 ∧ g; h2)

= a(i, j, k)e j,k(h1 ∧ g)ei (h2)+ a( j, i, k)ei,k(h1 ∧ g)e j (h2)

+ a(k, i, j)ei, j (h1 ∧ g)ek(h2)

=(a(i, j, k)(ek ⊗ e j − e j ⊗ ek)⊗ ei

+ a( j, i, k)(ek ⊗ ei − ei ⊗ ek)⊗ e j

+ a(k, i, j)(e j ⊗ ei − ei ⊗ e j )⊗ ek)(g; h1; h2),

(id⊗∂Hm)Ya(i, j, k)(g; h1; h2)

=(a(i, j, k)(ek ⊗ e j ⊗ ei − e j ⊗ ek ⊗ ei )

+ a( j, i, k)(ek ⊗ ei ⊗ e j − ei ⊗ ek ⊗ e j )

+ a(k, i, j)(e j ⊗ ei ⊗ ek − ei ⊗ e j ⊗ ek))(g; h1; h2)

= Xa(i, j, k)(h1 ∧ g; h2)

and (id⊗∂Hm)XAS a(i, j, k) = 0. Hence Lemma 4.3 again yields, for each g ∈ Land h1, h2 ∈ Hm,

(id⊗∂Hm)(Ya(i, j, k)+ XAS a(i, j, k))(g; h1; h2)

≡ (Ya(i, j, k)+ XAS a(i, j, k))(h1 ∧ g; h2) mod Z.

Similarly, we get ((id⊗∂Hm)Ya(i, k))(g, h1; h2) = Ya(i, k)(h1 ∧ g; h2) for g ∈ Land h1, h2 ∈ Hm, and XAS a(i, k) = 0. Thus so far we have established formulas(a) and (b).

Now we work on (c). Fixing g, h ∈ L , we compute its right hand side as

µa(h; h−1gh)µa(g; h)

=µa(h; (g∧ h)g)

µa(g; h)= λa(g∧ h; h)

µa(h; g)µa(g; h)

= λa(g∧ h; h)µa(m0(h)sH (h);m0(g)sH (g))µa(m0(g)sH (g);m0(h)sH (h))

= exp(2π i(Xa(g∧ h; h)))(exp(2π i(AS Va(h; g))))

= exp(2π i((Ya + XAS a)(g∧ h; h)))(exp(2π i(AS Va(h; g))).

Next we prove that

λa(sH (g); sH (h))= λa(g∧ h; h)(ASµa)(sH (h); sH (g)) for g, h ∈ N .

First we observe that

XAS a(g; h)≡ 0 mod Z for g, h ∈ L .


To prove (c), we ignore the term XAS a and compute

Xa(i, j, k)(g∧ h; h)

= a(i, j, k)e j,k(g∧ h)ei (h)+ a( j, i, k)ei,k(g∧ h)e j (h)

+ a(k, i, j)ei, j (g∧ h)ek(h)

=(a(i, j, k)(e j ⊗ (ekei )− ek ⊗ (e j ei ))

+ a( j, i, k)(ei ⊗ (eke j )− ek ⊗ (ei e j ))

+ a(k, i, j)(ei ⊗ (e j ek)− e j ⊗ (ei ek)))(g; h),

and also

Xa(i, k)(g∧ h; h)= a(i, i, k)(ei (g)ek(h)− ek(g)ei (h))ei (h)

+ a(k, i, k)(ei (g)ek(h)− ek(g)ei (h))ek(h)

= a(i, i, k)(ei ⊗ (ei ek)− ek ⊗ e2i )(g; h)

+ a(k, i, k)(ei ⊗ e2k − ek ⊗ (ei ek))(g; h).

Next we determine the asymmetrization of Ua(i, j, k) based on (4-18):

AS Ua(i, j, k)= 16(a(i, j, k)AS fi, j,k + a( j, i, k)AS f j,i,k + a(k, i, j)AS fk,i, j

− (AS a)(i, j, k)AS fi, j,k)

=12

(a(i, j, k)((e j ek)∧ ei − (ei ek)∧ e j + (ei e j )∧ ek)

+ a( j, i, k)((ei ek)∧ e j − (e j ek)∧ ei + (ei e j )∧ ek)

+ a(k, i, j)((ei e j )∧ ek − (e j ek)∧ ei + (ei ek)∧ e j )

− (a(i, j, k)− a( j, i, k)+ a(k, i, j))

× ((e j ek)∧ ei − (ei ek)∧ e j + (ei e j )∧ ek))

=12

(a( j, i, k)((ei ek)∧ e j − (e j ek)∧ ei + (ei e j )∧ ek)

+ a(k, i, j)((ei e j )∧ ek − (e j ek)∧ ei + (ei ek)∧ e j )

+ (a( j, i, k)− a(k, i, j))

× ((e j ek)∧ ei − (ei ek)∧ e j + (ei e j )∧ ek))

=−a(k, i, j)(e j ek)∧ ei + a(k, i, j)(ei ek)∧ e j

+ a( j, i, k)(ei e j )∧ ek .

Hence we get

(4-24) AS Ua(i, j, k)=−a(k, i, j)((e j ek)⊗ ei − ei ⊗ (e j ek))

+ a(k, i, j)((ei ek)⊗ e j − e j ⊗ (ei ek))

+ a( j, i, k)((ei e j )⊗ ek − ek ⊗ (ei e j )).


We also check the asymmetrization of Ua(i, k):

AS Ua(i, k)= a(i, i, k)ek ∧ Bi i + a(k, i, k)(Bkk ∧ ei − ek ∧ (ei ek))

=12a(i, i, k)(e2

i ⊗ ek − ek ⊗ e2i )+

12a(k, i, k)(ei ⊗ e2

k − e2k ⊗ ei )

+ a(k, i, k)((ei ek)⊗ ek − ek ⊗ (ei ek)).

We then combine these with the above computations for Xa(i, j, k), paying atten-tion to the order of variables in the first and second term:1

Xa(i, j, k)(g∧ h; h)+AS Ua(i, j, k)(sH (h); sH (g))

= a(i, j, k)(e j ⊗ (ekei )− ek ⊗ (e j ei ))

+ a( j, i, k)(ei ⊗ (eke j )− ek ⊗ (ei e j ))

+ a(k, i, j)(ei ⊗ (e j ek)− e j ⊗ (ei ek))

+(a(k, i, j)(e j ek)∧ ei − a(k, i, j)(ei ek)∧ e j

− a( j, i, k)(ei e j )∧ ek)


+ a( j, i, k)(ei ⊗ (eke j )− ek ⊗ (ei e j )− (ei e j )∧ ek

)+ a(k, i, j)

(ei ⊗ (e j ek)− e j ⊗ (ei ek)+ (e j ek)∧ ei − (ei ek)∧ e j

)= a(i, j, k)(e j ⊗ (ekei )− ek ⊗ (e j ei ))

+ a( j, i, k)(ei ⊗ (eke j )− ek ⊗ (ei e j )− (ei e j )⊗ ek + ek ⊗ (ei e j )

)+ a(k, i, j)

(ei ⊗ (e j ek)− e j ⊗ (ei ek)+ (e j ek)⊗ ei

− ei ⊗ (e j ek)− (ei ek)⊗ e j + e j ⊗ (ei ek))


+ a( j, i, k)(ei ⊗ (eke j )− (ei e j )⊗ ek)

+ a(k, i, j)((e j ek)⊗ ei − (ei ek)⊗ e j ).

and

Xa(i, k)(g∧ h; h)+AS Ua(i, k)(sH (h); sH (g))

= a(i, i, k)(ei ⊗ (ei ek)− ek ⊗ e2i )

+ a(k, i, k)(ei ⊗ e2k − ek ⊗ (ei ek))

+12a(i, i, k)(ek ⊗ e2

i − e2i ⊗ ek)+

12a(k, i, k)(e2

k ⊗ ei − ei ⊗ e2k)

+ a(k, i, k)(ek ⊗ (ei ek)− (ei ek)⊗ ek)

= a(i, i, k)(ei ⊗ (ei ek)−

12(ek ⊗ e2

i + e2i ⊗ ek)

)+ a(k, i, k)

( 12(ei ⊗ e2

k + e2k ⊗ ei )− (ei ek)⊗ ek

).

1In the first term, the variables g and h appear in this order, but in the second they appear in theopposite order.


We now compare these with Ya(i, j, k):

Ya(i, j, k)(sH (g); sH (h))

= a(i, j, k)(e j ⊗ (ei ek)− ek ⊗ (ei e j ))

+ a( j, i, k)(ei ⊗ (e j ek)− (ei e j )⊗ ek)

+ a(k, i, j)((e j ek)⊗ ei − (ekei )⊗ e j )

= Xa(i, j, k)(g∧ h; h)+AS Ua(i, j, k)(sH (h); sH (g))

≡ Ya(i, j, k)(g∧ h; h)+AS Ua(i, j, k)(sH (h); sH (g)),

and

Ya(i, k)(sH (g); sH (h))

=(a(i, i, k)(Bi i ⊗ ek + ek ⊗ Bi i − Bik ⊗ ei − ei ⊗ Bki )

+ a(k, i, k)(Bki ⊗ ek + ek ⊗ Bik − Bkk ⊗ ei − ei ⊗ Bkk))

= a(i, i, k)(ei ⊗ (ei ek)−

12(ek ⊗ e2

i + e2i ⊗ ek)

)+ a(k, i, k)

( 12(ei ⊗ e2

k + e2k ⊗ ei )− (ei ek)⊗ ek

)= Xa(i, k)(g∧ h; h)+AS Ua(i, k)(sH (h); sH (g))

= Ya(i, k)(g∧ h; h)+AS Ua(i, k)(sH (h); sH (g)).

Therefore, we have

λa,b(sH (g); sH (h))= λa,b(g∧ h; h)µa(sH (h); sH (g))µa(sH (g); sH (h))

.

Since we have Ya(mg; nh) = Ya(m; h)+ Ya(g; n)+ Ya(g; h) for every m, n ∈ Mand g, h ∈ Hm, we get, for each m, n ∈ M and g, h ∈ N ,

λa,b(msH (g); nsH (h))

= λa,b(m; sH (h))λa,b(sH (g); n)λa,b(sH (g); sH (h))

=λa,b(m; sH (h))λa,b(n; sH (g))

λa,b(g∧ h; h)µa(sH (h); sH (g))µa(sH (g); sH (h))

=µa(nsH (h); (nsH (h))−1msH (g)nsH (h))

µa(msH (g); nsH (h)).

This proves the cocycle identity (c). Consequently {λa,b, µa} is a characteristiccocycle in Z(Hm, L ,M,T).

(ii) Suppose that (λ, µ)∈Z(Hm, L ,M,T). Since M is central in Hm, the λ-partis a bicharacter on M × Hm, so there exists an a = {a(i, j, k)} ∈ R1 such that

λ(m; h)= exp(

2π i(∑

i, j<k

a(i, j, k)e j,k(m)ei (h)))

for m ∈ M and h ∈ Hm.


As [Hm, Hm] =M , for each fixed m ∈M the character λ(m; · ) on Hm must vanishon M , that is, λ(m; n) = 1 for m, n ∈ M . Thus the restriction µM of the secondcocycle µ to M is a coboundary. Hence, replacing µ by a cohomologous cocycleif necessary, we may and do assume that µM = 1. Now consider the correspondingE ∈ Xext(Hm, L ,M,T), with diagram

1→ T // Ej←−s j

// L→ 1.

Redefining the cross-section s j as s j (msH (g)) = s j (m)s j (sH (g)) for m ∈ M andg ∈ N , we may and do assume that µ(m; g) = 1 for m ∈ M and g ∈ L . Now wecompute the second cocycle µ with m, n ∈ M and g, h ∈ L:

µ(mg; nh)s j (mgnh)= s j (mg)s j (ng)= s j (m)s j (g)s j (n)s j (h)

= s j (m)λ(n; g)s j (n)s j (g)s j (h)

= λ(n; g)µ(g; h)s j (m)s j (n)s j (gh)

= λ(n; g)µ(g; h)s j (mngh)= λ(n; g)µ(g; h)s j (mgnh),

which gives µ(mg; nh)= λ(n; g)µ(g; h) for m, n ∈M and g, h ∈ L . In particular,we have

µ(g; h)= λ(m0(h); g)µ(sH (πG(g)); sH (πG(h)) for g, h ∈ L ,

where m0(h) = hsH (πG(h))−1∈ M . Now with g1, g2, g3 ∈ N , we compute the

coboundary:

(4-25)

1= (∂Lµ)(sH (g1); sH (g2); sH (g3))

=µ(sH (g2); sH (g3))µ(sH (g1); sH (g2)sH (g3))

µ(sH (g1)sH (g2); sH (g3))µ(sH (g1); sH (g2))

=µ(sH (g2); sH (g3))µ(sH (g1); nM(g2; g3)sH (g2+ g3))

µ(nM(g1; g2)sH (g1+ g2); sH (g3))µ(sH (g1); sH (g2))

= λ(nM(g2; g3); sH (g1))µ(sH (g2); sH (g3))µ(sH (g1); sH (g2+ g3))

µ(sH (g1+ g2); sH (g3))µ(sH (g1); sH (g2)).

Thus the cocycle ca ∈ Z3(N ,T) given by

ca(g1; g2; g3)= λ(nM(g2; g3); g1)

= exp(

2π i(∑

i, j<k

a(i, j, k)e j,k(nM(g2; g3))ei (g1)))

= exp(

2π i(∑

i, j<k

a(i, j, k)ei (g1)e j (g2)ek(g3)))


is a coboundary in Z3(N ,T). Thus we get, for every g1, g2, g3 ∈ N ,

1= (AS ca)(g1, g2, g3)

= exp(

2π i(∑

i, j<k

a(i, j, k)∑

σ∈5(i, j,k)

sign(σ )ei (gσ(i))e j (gσ( j))ek(gσ(k))))

= exp(

2π i(∑

i, j<k

a(i, j, k) deti jk(g1; g2; g3)))

= exp(

2π i( ∑(i, j,k)∈1

(AS a)(i, j, k) deti jk(g1, g2, g3))).

Thus the coefficient a={a(i, j, k)}∈R1 is asymmetric in the sense of Lemma 4.4,so that it gives the second cocycle µa = exp(2π iVa) ∈ Z2(L ,T). Then the cocycleµµ−1

a ∈ Z2(L ,T) falls in the subgroup π∗G(Z2(N ,T))⊂ B2(L ,T) because

µ(msH (g); nsH (h))= λ(n; sH (g))µ(sH (g); sH (h))

=µa(msH (g); nsH (h))µa(sH (g); sH (h))

µ(sH (g); sH (h))

=µ(sH (g); sH (h))µa(sH (g); sH (h))

µa(msH (g); nsH (h)),

µ−1a µ= π∗G ◦ s

∗

H (µµ−1a ) ∈ π∗G(Z

2(N ,T)).

Thus there exists a cochain f ∈ C1(L ,T) such that

µa(g; h)= µ(g; h)f (g) f (h)

f (gh)for g, h ∈ L .

Since 1=µ(m; h)=µa(m; h) for m ∈M and h ∈ L , we have f (mh)= f (m) f (h).Since (∂1 f )(m; h)= 1 for m ∈ M and h ∈ Hm, we have ∂ f (λ, µ)= (λ, µa).

Next we look at one of the cocycle identities, for g1, g2 ∈ L and h ∈ Hm:

λ(g1g2; h)= λ(g1; h)λ(g2; h)µa(g1; g2)

µa(h−1g1h; h−1g2h)

=1

λ(g2∧h; g1)λ(g1; h)λ(g2; h)

= λ(h ∧ g2; g1)λ(g1; h)λ(g2; h)

= exp(

2π i(∑

i, j<k

a(i, j, k)ei (g1)e j,k(h ∧ g2)))λ(g1; h)λ(g2; h),

which gives the partial coboundary condition

(∂L ⊗ id)λ= exp(

2π i(∑

i, j<k

a(i, j, k)ei ⊗ (e j ⊗ ek − ek ⊗ e j ))).


Another cocycle identity for g ∈ L and h1, h2 ∈ Hm is

λ(g; h1h2)= λ(g; h1)λ(h−11 gh1; h2),

= λ(g∧ h1; h2)λ(g; h1)λ(g; h2)

= exp(

2π i(∑

i, j<k

a(i, j, k)e j,k(g∧ h1)ei (h2)))λ(g; h1)λ(g; h2);

this gives the second partial coboundary condition

(id⊗∂Hm)λ= exp(

2π i(∑

i, j<k

a(i, j, k)(ek ⊗ e j − e j ⊗ ek)⊗ ei

)).

Setting ηa = exp(2π i(Ya)), we obtain, by (4-22) and (4-23),

(∂L ⊗ id)λ= (∂L ⊗ id)ηa and (id⊗∂Hm)λ= (id⊗∂Hm)ηa.

Therefore the cochain ηaλ= χ is a bicharacter on L×Hm. Since M = [Hm, Hm],the bicharacter χ vanishes on L ×M , that is, λ(m; g) = ηa(m; g) for m ∈ M andg ∈ L . Thus we get

1= λ(m; g)ηa(m; g)= exp(2π i(Xa(m; g)− Ya(m; g)))

= exp(2π i(XAS a(m; g)))= λAS a(m; g),

which is equivalent to the fact that (AS a)(i, j, k) ∈ ((1/gcd(pi , p j , pk))Z). Thuswe conclude the cocycle condition (4-9Z-a) on the parameter {a(i, j, k)}. There-fore the coefficient a ∈ R1 satisfies the requirement for the element (a, 0) ∈ Z .Consequently, it follows from (i) that (λa,0, µa) ∈ Z(Hm, L ,M,T). Then the co-cycle identity (c) for (λa,0, µa) yields that

λ(g; h)=µa(h; h−1gh)µa(g; h)

= λa,0(g; h)= ηa(g; h) for g, h ∈ L .

Thus the bicharacter χ = ηaλ on L× Hm vanishes on L× L . Since Lemma 4.3(i)yields for each m ∈ M and h ∈ Hm that

χ(m; h)= λ(m; h)ηa(m; h)= λa(m; h)ηa(m; h)

= λAS a(m; h)= exp(

2π i(∑

i, j<k

(AS a)(i, j, k)e j,k(m)ei (h))),

we conclude that χ is of the form

χ(g; h)= χ0(πG(g);πG(h)) exp(

2π i( ∑

i< j<k

(AS a)(i, j, k)e j,k(g)ei (h)))


for g ∈ L and h ∈ Hm, where χ0 is a bicharacter on N ×Gm and πG : Hm→ Gm

the quotient map with M = Ker(πG). We choose b(i, j) ∈ R so that

exp(2π i(b(i, j)))= χ0(bi ; z j ) for i ∈ N and j ∈ N0.

Then we must have

1= χ0(bi ; b j )= χ0(bi ; p j z j − q j z0)= exp(2π i(b(i, j)p j − b(i, 0)q j )),

so that b(i, j) ∈ R for i ∈ N and j ∈ N0 satisfies the condition

b(i, j)p j ≡ b(i, 0)q j mod Z for i, j ∈ N.

Hence χ0 is written in the form

χ0(g; h)= exp(

2π i(∑

i, j∈N

b(i, j)ei,N (g)e j (h)+∑i∈N

b(i, 0)ei,N (g)e0(h)))

for each pair g ∈ N and h ∈ Hm, where each coefficient b(i, j) satisfies

b(i, j)p j − b(i, 0)q j ∈ Z for i, j ∈ N and b(0, i)= 0 for i ∈ N0.

Consequently the pair (a, b) is a member of Z and we conclude that (λ, µ) iscohomologous to the characteristic cocycle (λa,b, µa) ∈ Z(Hm, L ,M,T).

(iii) Suppose (λ, µ) = (λa,b, µa) = ∂ f with f ∈ C1(L ,T). Since µM = 1 andµa(m; g)= 1 for m ∈ M and g ∈ L , we have f (mg)= f (m) f (g) for m ∈ M andg ∈ L , so that the restriction of f to M is of the form

fc(m)= exp(

2π i( ∑

1≤i< j

c(i, j)ei, j (m)))

for m ∈ M.

Since M is central in Hm, we have for every pair (m, g) ∈ M × Hm

1=fc(g−1mg)

fc(m)= λ(m; g)= exp

(2π i

(∑i, j<k

a(i, j, k)e j,k(m)ei (g))),

which yields the integrality condition a(i, j, k) ∈ Z for every (i, j, k) ∈ 1. thatλa(m; h) = 1 for m ∈ M and h ∈ Hm. Since χ = 1 on L × L , for every g, h ∈ Lwe have

1= λ0,b(g; h)= λa(m0(g); h)λ0,b(sH (g); h)= λ(g; h)

= f (h−1gh)/ f (g)= fc(g∧ h);

c(i, j) ∈( 1

pi p jZ)

for i, j ∈ N.

This computation also shows that

λ0,b(g; h)= fc(g∧ h) for g ∈ L and h ∈ Hm.


Furthermore, we have for each m, n ∈ M and g, h ∈ L

µa(mg; nh)= λa,b(n; g)µa(g; h)= µa(g; h),

so that µa is of the form µa = π∗G(µ) with

µa(g; h)= exp(2π i(Ua(g; h))) for g, h ∈ N .

Since λa(nM(g2; g3); g1)=1 for g1, g2, g3∈Hm, we have µa ∈Z2(N ,T) by (4-25).We first compute for each g, h ∈ L that

(ASµa)(g; h)=f (g) f (h)

f (gh)f (hg)

f (g) f (h)=

f (hgh−1h)f (gh)

= fc(h ∧ g)= 1.

Since AS Ua(i, j, k) is also integer valued, we have

ASµa = exp(

2π i(∑

i<k

AS Ua(i, k)))

= exp(

2π i(∑

i<k

(12a(i, i, k)(e2

i ⊗ ek − ek ⊗ e2i ))))

× exp(

2π i(∑

i<k

12a(k, i, k)(ei ⊗ e2

k − e2k ⊗ ei )

))= 1.

Thus we get

a(i, i, k), a(k, i, k) ∈ 2Z and Ua(i, k)≡ 0 mod Z.

Consequently, µa is a coboundary as a member of Z2(N ,T). Hence there exists acochain f ∈ C1(N ,T) such that

f (g) f (h)f (gh)

= µa(g; h)= µa(πG(g);πG(h))=f (πG(g)) f (πG(h))

f (πG(gh)).

Thus f is of the form

f (g)= fc(m0(g)) f (sH (g))= χ(g) f (πG(g)) for g ∈ L ,

fc(m)= χ(m) for m ∈ M.

where χ ∈ Hom(L ,T). Since L/[L , L] ∼= M/P M P ⊕ N , the homomorphism χ

is of the form

χ(g)= exp(

2π i(∑

j<k

c( j, k)e j,k(g)+∑k∈N0

c(k)ek(g)))

for g ∈ L ,

wherec(i, j) ∈

( 1pi p j

Z)

for i < j and c(k) ∈ R.


Since Ya is integer valued, the λ part becomes for g ∈ N and h ∈ Hm

λ(g; h)= exp(

2π i( ∑

j∈N,k∈N0

b( j, k)e j,N (g)ek(h))),

=f (h−1gh)

f (g)=

f ((g∧ h)g)f (g)

= fc(g∧ h)

= exp(

2π i( ∑

1≤ j<k

c( j, k)e j,k(g∧ h)))

= exp(

2π i( ∑

1≤ j<k

c( j, k)(e j (g)ek(h)− ek(g)e j (h)

)))= exp

(2π i

( ∑1≤ j<k

c( j, k)(

p j e j,N (g)ek(h)− pkek,N (g)e j (h))))

.

Hence we conclude that for j < k and i ∈ N

b(i, 0) ∈ Z, b( j, k)≡ c( j, k)p j mod Z,

b(i, i) ∈ Z, b(k, j)≡−c( j, k)pk mod Z.

Thus we have for i < j

b(i, j)= c(i, j)pi +mi, j for some mi, j ∈ Z,

b( j, i)=−c(i, j)p j +m j,i for some m j,i ∈ Z,

b(i, j)pi+

b( j, i)p j=

mi, j

pi+

m j,i

p j∈

( 1pi

Z)+

( 1p j

Z)=

( 1lcm(pi , p j )

Z).

Conversely suppose (a, b) ∈ B, that is,

a(i, j, k) ∈ Z for i < j < k and a(i, i, k), a(k, i, k) ∈ 2Z for i < k,

and b(i, j)/pi + b( j, i)/p j ∈ ((1/lcm(pi , p j ))Z); also b(i, i) ∈ Z and b(i, 0) ∈ Z

for i ∈ N. So we can write

b(i, j)pi+

b( j, i)p j=

mi, j

pi+

m j,i

p jfor some mi, j ,m j,i ∈ Z.

Set c(i, j)= b(i, j)/pi −mi, j/pi for i < j and c(i, i)= b(i, i), so that

b( j, i)p j=−c(i, j)+

m j,i

p j.


Then we have∑i, j∈N

b(i, j)ei,N (g)e j (h)

≡

∑i< j

c(i, j)(

pi ei,N (g)e j (h)− p j e j,N (g)ei (h))

mod Z,

=

∑i< j

c(i, j)ei, j (g∧ h).

Thus with fc(g)= exp(2π i

(∑1≤i< j c(i, j)ei, j (g)

))for g ∈ L , we have

exp(

2π i(∑

i, j

b(i, j)ei,N (g)e j (h)))=

fc(h−1gh)fc(g)

= ∂1 fc(g; h),

where ei,N (g)means ei,N ◦πG . We then compute the coboundary of fc for g, h ∈ Las

(∂L fc)(g; h)=fc(g) fc(h)

fc(gh)

= exp(

2π i(∑

i< j

c(i, j)(ei, j (g)+ ei, j (h)− ei, j (gh))))

= exp(−2π i

(∑i< j

c(i, j)ei (g)e j (h)))= 1,

because ei (g) ∈ pi Z and e j (h) ∈ p j Z if g, h ∈ L and

pi c(i, j)p j = b(i, j)p j −mi, j p j ≡ b(i, 0)q j ≡ 0 mod Z.

As a(i, j, k) ∈ Z for every triplet (i, j, k) ∈1, we get trivially

λa,0 = 1, µa = s∗Hµa ∈ Z2(N ,T), and µa = π∗

G(µa).

Since ∂NUa(i, j, k) for i < j < k is integer valued, the cochain

µi jka = exp(2π i(Ua(i, j, k)))

belongs to Z2(N ,T). Since AS Ua(i, j, k) is integer valued by (4-24), AS µi jka = 1

and therefore µi jka ∈ B2(N ,T). Because µik

a = exp(2π i(Ua(i, k))) = 1 for i < k,we conclude that µa ∈ B2(N ,T). Thus there exists a cochain f ∈ C1(N ,T) suchthat µa = ∂N f . Define a cochain f ∈ C1(L ,T) by f = (π∗G f ) fc. Then we get foreach pair g ∈ L and h ∈ Hm

(∂1 f )(g; h)=f (h−1gh)

f (g)=

f (πG(h−1gh)) fc(h−1gh)

f (πG(g)) fc(g)=

fc(h−1gh)fc(g)

= λa,b(g; h)


and for g, h ∈ L

(∂2 f )(g; h)=f (πG(g)) fc(g) f (πG(h)) fc(h)

f (πG(gh)) fc(gh)

= ∂L fc(g; h)(∂N f )(πG(g);πG(h))= µa(πG(g);πG(h))= µa(g; h).

Therefore we conclude ∂ f = {λa,b, µa} ∈ B(Hm, L ,M,T). ♥

Lemma 4.6. The cocycle λb corresponding to b ∈ Zb does not depend on the M-component, that is,

λb(mg; nh)= λb(g; h) for m, n ∈ M, g ∈ L and h ∈ Hm.

We will view λb as a bicharacter on N ×Gm rather than on L × Hm.

(i) For i ∈ Z, set

Zb(i, i)= {z = (x, u) ∈ R2: pi x − qi u ∈ Z} and Bb(i, i)= Z⊕Z.

The bicharacter λi,iz on N ×Gm determined by

λi,iz (g; h)= exp(2π i(xei,N (g)ei (h)+ uei,N (g)e0(h))) for g ∈ N and h ∈ Gm,

gives a characteristic cocycle of Z(Hm, L ,M,T). It is a coboundary if andonly if z is in Bb(i, i). The corresponding cohomology class [λi,i

z ] ∈ 3b(i, i)is given by

[λi,iz ] = ([pi x − qi u]gcd(pi ,qi ), [−vi x + ui u]Z) ∈ Zgcd(pi ,qi )⊕ (R/Z),

where the integers ui and vi are determined by pi ui − qivi = gcd(pi , qi )

through the Euclid algorithm.

(ii) Fix a pair i, j ∈ N of indices and set

Zb(i, j)= {(x, u, y, v) ∈ R4: p j x − q j u ∈ Z, pi y− qiv ∈ Z},

Bb(i, j)= {(x, u, y, v) ∈ Zb(i, j) : p j x + pi y ∈ gcd(pi , p j )Z, u, v ∈ Z}.

To each element z = (x, u, y, v) ∈ Zb(i, j), there corresponds a bicharacterλz on N ×Gm determined by

λi, jz (g; h)= exp(2π i(xei,N (g)e j (h)+ ye j,N (g)ei (h)))

× exp(2π i(uei,N (g)e0(h)+ ve j,N (g)e0(h))) for g ∈ N and h ∈ Gm,

which is a characteristic cocycle in Z(Hm, L ,M,T). It is a coboundary if andonly if z ∈ Bb(i, j). The cohomology class [λi, j

z ] ∈3b(i, j) of λz corresponds


to the parameter class

[z] =



∈( 1

D(i, j)Z)/

Z

R/Z

R/Z

,where D(i, j), . . . , w j,i are given in (4-17) of Theorem 4.2.

Proof. (i) Set Di = gcd(pi , qi ); set ri = pi/Di and si = qi/Di , and choose integersui , vi ∈Z so that ri ui−sivi = 1, where such a pair (ui , vi )∈Z2 can be determinedthrough the Euclid algorithm. Next we set e1 = (1, 0) and e2 = (0, 1). Set

f1 = ui e1+ vi e2 and f2 = si e1+ ri e2,

so that e1 = ri f1− vi f2 and e2 =−si f1+ ui f2. Then Zb(i, i) is given by

Zb(i, i)=( 1

DiZ)

f1+R f2,

and Bb(i, i)= Ze1+Ze2 = Z f1+Z f2, so that

3b(i, i)= Zb(i, i)/Bb(i, i)∼=( 1

DiZ/

Z)

f1⊕ (R/Z) f2,

where the dotted elements indicate the corresponding elements in the quotientgroup 3b(i, i). Now we chase the parameter:

z = xe1+ ue2 = x(ri f1− vi f2)+ u(−si f1+ ui f2)

= (ri x − si u) f1+ (−vi x + ui u) f2;

z = [ri x − si u]Z f1+ [−vi x + ui u]Z f2,

and

λi,iz (g; h)= exp(2π i((xei,N (g)ei (h)+ uei,N (g)e0(h))))

for each pair g ∈ N and h ∈ Gm.(ii) First we fix the standard basis {e1, . . . , e4} of R4 and set

g0 = ri, j e1− r j,i e3 and g1 = u j,i e1+ ui, j e3,

where we choose ui, j , u j,i ∈ Z so that ri, j ui, j + r j,i u j,i = 1. Since

e1 = ui, j g0+ r j,i g1 and e2 =−u j,i g0+ ri, j g1,

we have Ze1+Ze3 = Zg0+Zg1. Also we have

Bb(i, j)+Rg0 = Rg0+Zg1+Ze2+Ze4.


Consider an integer 3× 4 matrix

T =

mi, jr j,i −ni, j s j,i mi, jri, j −ni, j si, j

yi, jr j,i xi, j s j,i yi, jri, j xi, j si, j

0 −wi, j 0 w j,i

.We claim that

T (Zb(i, j)+Rg0)=( 1

D(i, j)Z)⊕R⊕R.

To prove the claim, for each vector z = xe1 + ue2 + ye3 + ve4 ∈ R4, we simplycompute

T g0 = 0,

T z =

mi, jr j,i −ni, j s j,i mi, jri, j −ni, j si, j

yi, jr j,i xi, j s j,i yi, jri, j xi, j si, j

0 −wi, j 0 w j,i

xuyv

=

mi, j (xr j,i + yri, j )− ni, j (us j,i + vsi, j )

yi, j (xr j,i + yri, j )+ xi, j (us j,i + vsi, j )

−uwi, j + vw j,i

.Suppose

kD(i, j)

= mi, j (xr j,i + yri, j )− ni, j (us j,i + vsi, j ) ∈( 1

D(i, j)

)Z.

Then we have

k = (mi, j (xr j,i + yri, j )− ni, j (us j,i + vsi, j ))D(i, j)

= (x p j − uq j )+ (y pi − vqi )

= ((x + tri, j )p j − uq j )+ ((y− tr j,i )pi − vqi ).

A choice of t ∈R, such that (x+ tri, j )p j −uq j is an integer, yields the integralityof the other term (y− tr j,i )pi −vqi , so that z+ tg0 ∈ Zb(i, j). Now we prove that

T−1Z3= Bb(i, j)+Rg0.

Since T is a matrix with integer coefficients and the generators g1, e2, e4 are allinteger vectors, we have T (Bb(i, j))⊂Z3. Conversely, suppose that T z ∈Z3. Thenwe have

k = mi, j (xr j,i + yri, j )− ni, j (us j,i + vsi, j ) ∈ Z,

`= yi, j (xr j,i + yri, j )+ xi, j (us j,i + vsi, j ) ∈ Z,

m =−uwi, j + vw j,i ∈ Z.


Hence we get

xr j,i + yri, j = xi, j k+ ni, j` ∈ Z, n = us j,i + vsi, j =−yi, j k+mi, j` ∈ Z,

u = nw j,i −msi, j ∈ Z, v = nwi, j +ms j,i ∈ Z,

x p j + y pi = (xr j,i + yri, j )Di, j ∈ Di, j Z.

Therefore z ∈ Bb(i, j)+Rg0.Consequently, we conclude

3b(i, j)∼= Zb(i, j)/Bb(i, j)∼=(( 1

D(i, j)Z)/

Z)⊕ (R/Z)⊕ (R/Z),

in the sense that the cohomology class [λi, jz ] ∈3b(i, j) corresponds to

[z] =



∈((1/D(i, j))Z)/Z

R/Z

R/Z

.For each i, j ∈ N, define maps π i

i : 3b(i, i)→ R/Z, π ii, j : 3b(i, j)→ R/Z,

πj

i, j :3b(i, j)→ R/Z and πi j :3b(i, j)→ ((1/D(i, j))Z)/Z by

π ii ([λ

i,iz ])= [u]Z ∈ R/Z and πi i ([λ

i,iz ])= [xri − usi ]Z ∈ ((1/Di )Z)/Z

for each z = (x, u) ∈ Zb(i, i), and

π ii, j ([λ

i, jz ])= [u]Z ∈ R/Z, π

ji, j ([λ

i, jz ])= [v]Z ∈ R/Z,

πi j ([λi, jz ])= [mi, j (xr j,i + yri, j )− ni, j (us j,i + vsi, j )]Z ∈

( 1D(i, j)

Z)/

Z

for each z = (x, u, y, v) ∈ Zb(i, j). The maps π ii, j and π j

i, j are both well definedbecause the coboundary condition on z implies the integrality of u and v.

Let 3b be the set of all

λb = {λb(i, i), λb(i, j)} ∈∏i∈N

3b(i, i)×∏i< j

i, j∈N

3b(i, j)

such that π ii (λb(i, i))=π i

i, j (λb(i, j))=π iki (λb(k, i)) for all i, j, k ∈N. Finally we

have 3(Hm, L ,M,T)=3a ⊕3b. This completes the proof. ♥

Remark 4.7. The direct sum homomorphism πi j⊕πii, j⊕π

ji, j is a homomorphism

of 3a(i, j) onto the direct sum group:

3b(i, j)πi j⊕π

ii, j⊕π

ji, j //(( 1

D(i, j)Z)/

Z)⊕ (R/Z)⊕ (R/Z).


By multiplying πi i (λz) by Di , we get

Diπi i ([λz])= [x pi − uqi ]Di Z ∈ Z/(Di Z).

Similarly, we have

D(i, j)πi j (λz)= [(x p j + y pi )− (uq j + vqi )]D(i, j) ∈ Z/(D(i, j)Z).

The kernel of πi j ⊕πii, j ⊕π

ji, j is given by

Ker(πi j ⊕πii, j ⊕π

ji, j )=

{0}( 1

mi, jZ)/

Z

{0}

.At the parameter level, the kernel is described as follows:

[λz] ∈ Ker(πi j ⊕πii, j ⊕π

ji, j ) if and only if x p j + y pi ∈ D(i, j)Z, u, v ∈ Z.

5. The reduced modified HJR-sequence

We are now going to investigate the reduced modified HJR-exact sequence

(5-1)

...

��

...

��H2(H,T)

Res��

H2(H,T)

res��

3(Hm, L ,M,T)

δ

��

res // 3(H,M,T)

δHJR��

Houtm,s(G, N ,T)

Inf��

∂Qm // H3(G,T)

inf��

H3(H,T) H3(H,T)

We refer to [Katayama and Takesaki 2007, page 116] for details. We first discussthe second cohomology group Z2(H,T) and the restriction map Res. Each secondcocycle µ∈Z2(H,T) gives rise to a group extension equipped with a cross-section

1→ T // Ej←−s j

// H → 1


such that s j (g)s j (h)= µ(g; h)s j (gh) for g, h ∈ H . With

λµ(g; h)= µ(h; h−1gh)/µ(g; h) for g, h ∈ H,

we obtain a characteristic cocycle (λµ, µ) ∈ Z(H, H,T). This corresponds to thecase that P = 1 in the previous section. So we set

(5-2)Z2={a ∈ RN3

: a(i, j, k)= 0 if j ≥ k, (AS a)(i, j, k) ∈ Z},

B2={a ∈ Z2

: a(i, j, k) ∈ Z, a(i, i, k), a(k, i, k) ∈ 2Z}.

Theorem 5.1. (i) Each element a ∈ Z2 gives rise to a cocycle

(5-3) µa = exp(2π iVa) ∈ Z2(H,T)

and the diagram

1 // B2

��

// a ∈ Z2

��

// [a] ∈ H2 // 1

1 // B2(H,T) // µa ∈ Z2(H,T) // [λa] ∈ H2(H,T) // 1

describes the second cohomology H2(H,T). More precisely, with

Z2(i, j, k)= {(x, y, z) ∈ R3: x − y+ z ∈ Z}, Z2(i, k)= R2,

B2(i, j, k)= Z3, B2(i, k)= (2Z)2,

H2(i, j, k)= Z2(i, j, k)/B2(i, j, k), H2(i, k)= Z2(i, k)/B2(i, k)

for each triplet i < j < k (respectively pair i < k) and

a(i, j, k)= x, a( j, i, k)= y, a(k, i, j)= z,

(respectively a(i, i, k)= x, a(k, i, k)= y),

we setµi jk

a = exp(2π i(Va(i, j, k))) ∈ Z2(H,T),

µika = exp(2π i(Va(i, k))) ∈ Z2(H,T).

Then we have

Z2(H,T)=∏

i< j<k

Z2(i, j, k)×∏i<k

Z2(i, k),

B2(H,T)=∏

i< j<k

B2(i, j, k)×∏i<k

B2(i, k),

µa =

( ∏i< j<k

µi jka

)(∏i<k

µika

)∈ Z2(H,T),


H2(H,T)∼=∏

i< j<k

H2(i, j, k)×∏i<k

H2(i, k),

[µa] = ([µi jka ], [µ

ika ] : i < j < k and i < k) ∈ H2(H,T).

Each H2(i, j, k) for i < j < k, (respectively H2(i, k) for i < k), is given by

H2(i, j, k)∼= (R/Z)⊕ (R/Z),

(respectively H2(i, k)∼= (R/2Z)⊕ (R/2Z)).

Proof. Most of the claims have been proved already except the claim for the struc-ture of H2(i, j, k). To prove this, it is convenient to introduce a matrix

A =

1 −1 10 1 00 0 1

∈ SL(3,Z), for which A−1=

1 1 −10 1 00 0 1

.We then observe that AZ2(i, j, k)= (Z⊕R⊕R) and AB2

= Z3; we conclude

H2(i, j, k)∼= {0}⊕ (R/Z)⊕ (R/Z). ♥

Theorem 5.2. (i) Each second cocycle µa ∈ Z2(H,T) for a ∈ Z2 gives the cor-responding characteristic cocycle

Res(µa)= (λa, µa)= π∗

m(λa|L×Hm, µa|L) ∈ Z(Hm, L ,M,T).

The image Res(Z2(H,T)) is therefore given by

Res(Z2(H,T))= {(λa, µa) : a ∈ Za, (AS a)(i, j, k) ∈ Z, i < j < k}.

The (i, j, k)-component Res(i, j, k) of the restriction map Res gives rise tothe following commutative diagram of short exact sequences:

1

��

1

��B2(i, j, k)= Z3

��

Xa(i, j,k)−→Xa(i, j,k) // Ba(i, j, k)= Z3

��Z2(i, j, k)= A−1(Z⊕R2)

��

Xa(i, j,k)−→Xa(i, j,k)// Za(i, j, k)= A−1((1/D)Z⊕R2)

��H2(i, j, k)= {0}⊕T2

��

Res(i, j,k) // 3a(i, j, k)= ZD ⊕T2

��1 0,


where D = D(i, j, k)= gcd(pi , p j , pk). Also the restriction map Resa(i, k) :H2(i, k)→3a(i, k) is given by

1

��

1

��B2(i, k)= (2Z)2

��

Xa(i,k)−→Xa(i,k) // Ba(i, k)= (2Z)2

��Z2(i, k)= R2

��

Xa(i,k)−→Xa(i,k) // Za(i, k)= R2

��H2(i, k)= (R/2Z)2

��

Res(i,k) // 3a(i, k)= (R/2Z)2

��1 1

Consequently, we get

3a(i, j, k)/Res(i, j, k)(H2(i, j, k))∼= Z/(DZ),

3a(i, k)/Res(i, k)(H2(i, k))∼= {0}.

(ii) The modified HJR-map δ : 3(Hm, L ,M,T)→ Houtm,s(G, N ,T) enjoys these

properties:(a) The (i, j, k)-component and (i, k)-component of Ker(δ) are given by

Ker(δ)i jk = {0}⊕ (R/Z)⊕ (R/Z),

Ker(δ)ik = (R/2Z)⊕ (R/2Z)=3a(i, k).

(b) The image δ([λa, µa]) ∈ Houtm,s(G, N ,T) for a ∈ Za depends only on the

asymmetrization AS a, that is,

δ([λa, µa])= δ([λa, 1]),

where

(5-4)a(i, j, k)= (AS a)(i, j, k) ∈ ((1/D)Z) for i < j < k,

a( j, i, k)= a(k, i, j)= a(i, i, k)= a(i, j, j)= a(k, i, k)= 0.

(c) Set Za = {a ∈ Za : a satisfies the requirement (5-4)}. If a ∈ Za , then theimage ca = δ(λa, 1) ∈ Zout(Gm, N ,T) under the modified HJR-map δ isin the pull back π∗m(H

3(Q,T)) and given by

(5-5) ca(q1, q2, q3)= ca(q1, q2, q3)

= exp(

2π i( ∑

i< j<k

a(i, j, k){ei (q1)}pi {e j (q2)}p j {ek(q3)}pk

))for each q1 = (q1, s1), q2 = (q2, s2) and q3 = (q3, s3) ∈ Qm.


(d) The modified HJR-map δHJR is injective on3b and Ker(δ) is precisely theconnected component of 3(Hm, L ,M,T). If b ∈ Zb, then

[cb, νb] = δ(λb, 1) ∈ Zoutm,s(G, N ,T)

is given by

(5-6) cb(q1, q2, q3)= exp(

2π i( ∑

i∈N, j∈N0

b(i, j)ei,N (nN(q2; q3))e j (s(q1))))

where

(5-7)

ei,N (nN(q2; q3))= ηpi ([ei (q2)]pi ; [ei (q3)]pi )/pi ,

ei (s(q1))= {ei (q1)}pi for i ≥ 1,

e0(s(q1))= e0(q1).

The d-part dcb of cb is given by νb:

(5-8)

dcb(q2; q3)= exp(

2π i(∑

j∈N

b( j, 0)ηp j ([e j (q2)]p j ; [e j (q3)]p j )/p j

))= exp

(2π i

({νb(nN(q2; q3))}T /T

)),

νb(g)= πT

(T∑j∈N

b( j, 0)e j,N (g))∈ R/T Z for g ∈ N ,

where πT : s ∈ R 7→ sT = s+ T Z ∈ R/T Z is the quotient map.

The modular obstruction group Houtm,s(G, N ,T) looks like

(5-9) Houtm,s(G, N ,T)= Hout

a ⊕Houtb and Hout

b∼=3b,

δ([λa, µa])= [cAS a] ∈∏

i< j<k

(( 1gcd(pi , p j , pk)

Z)/

Z)

for a ∈ Za,

[cb, νb] = δ([λb, 1]) for νb ∈ Hom(N ,R/T Z),

[ci,ib ] = ([pi b(i, i)− qi b(i, 0)]Di Z, [−vi b(i, i)+ ui b(i, 0)]Z)

∈ Z/(Di Z)⊕R/Z,

[ci, jb ] =

[mi, j (b(i, j)r j,i + b( j, i)ri, j )− ni, j (b(i, 0)s j,i + b( j, 0)si, j )]Z[yi, j (b(i, j)r j,i + b( j, i)ri, j )+ xi, j (b(i, 0)s j,i + b( j, 0)si, j )]Z

[−b(i, 0)wi, j + b( j, 0)w j,i ]Z

∈

((1/D(i, j))Z)/ZR/Z

R/Z

, where D(i, j)= gcd(pi , p j , qi , q j ).


(iii) The map ∂Qm : Houtm,s(G, N ,T) → H3(G,T) in the modified HJR-exact se-

quence above is given by

(5-10)

∂Qm([ca][cbνb])= [cGa ] ∈ H3(G,T)= X3(G,T) for a ∈ Za,

where cGa = exp

(2π i

( ∑i< j<k

(AS a)(i, j, k)ei ⊗ e j ⊗ ek

)),

∂Qm(Houtm,s(G, N ,T))= π∗Q(H

3(Q,T)).

Proof. (i) The assertion has been already proved.(ii) For each i < j < k, let D(i, j, k)= gcd(pi , p j , pk) ∈ Z. Fix a ∈ Za , that is,

a ∈ R1 such that

(AS a)(i, j, k)= a(i, j, k)− a( j, i, k)+ a(k, i, j) ∈ ((1/D(i, j, k))Z),

a(i, j, k)= 0 if j ≥ k.

Set

za(i, j, k)=

a(i, j, k)a( j, i, k)a(k, i, j)

∈ Za = A−1

((1/D(i, j, k))Z)R

R

.Then we get

Aza(i, j, k)=

(AS a)(i, j, k)a( j, i, k)a(k, i, j)

∈(1/D(i, j, k))Z

R

R

ABa(i, j, k)= Z3,

so that

[λi, j,ka , µi jk

a ] ∼

[(AS a)(i, j, k)]Z[a( j, i, k)]Z[a(k, i, j)]Z

∈((1/D(i, j, k))Z)

R/Z

R/Z

.If (AS a)(i, j, k) ∈ Z, the second cocycle µi jk

a extends to a second cocycle on H ,which gives (λi, j,k

a , µi, j,ka ) = Res(µi, j,k

a ). Since Range(Res) = Ker(δ), the imageδ(λ

i, j,ka , µ

i, j,ka ) depends only on the first term (AS a)(i, j, k) of Aza(i, j, k). Hence

we conclude δ([λa, µa])= δ([λa], 1). We also have3a(i, k)=Res(i, k)(H2(i, k)),so that the map δ kills the entire 3a(i, k). This proves (ii)(a) and (ii)(b).

(ii)(c) Set ca = δ(λa, µa) with a ∈ Za . We then look at the crossed extensionEλa,µa ∈ Xext(Hm, L ,M,T), given by

1→ T // Ej←−s j

// L→ 1.


Since

a(i, j, k) ∈( 1

gcd(pi , p j , pk)Z)

and ei (g) ∈ pi Z for g ∈ L ,

we have µa = 1. Hence observing that λa(g; h) = 1 for every g ∈ L ∧ Hm andh ∈ Hm, we get from (3-15) and (3-16) that

ca(q1, q2, q3)= αs(q1)(s j (nL(q2; q3)))s j (nL(q1; q2q3))

×{s j (nL(q1; q2))s j (nL(q1q2; q3))}−1

= λa(s(q1)nL(q2; q3)s(q1)−1; s(q1))

= λa((s(q1)∧ nL(q2; q3))nL(q2; q3); s(q1))

= λa(s(q1)∧ nL(q2; q3); s(q1))λa(nL(q2; q3); s(q1))

= λa(nL(q2; q3); s(q1))

= exp(

2π i( ∑

i< j<k

a(i, j, k)e j,k(nL(q2; q3))ei (s(q1))))

= exp(

2π i( ∑

i< j<k


))= exp

(2π i

( ∑i< j<k


))= ca(q1; q2; q3)

for each q1 = (q1, s1), q2 = (q2, s2) and q3 = (q3, s3) ∈ Qm. The assertion (ii)(c)follows.

(ii)(d) Since Res(H2(H,T)) ∩3b = {0}, the modified HJR-map δ is injectiveon 3b. Now fix b ∈ Zb. Since µb = 1 and λb(m; h)= 1 for every pair m ∈ M andh ∈ Hm, we have, as in (ii)(c),

cb(q1; q2; q3)= λb(nN(q2; q3); s(q1))

= exp(

2π i( ∑

i∈N, j∈N0

b(i, j)ei,N (nN(q2; q3))e j (s(q1))))

= exp(

2π i(∑

i, j∈N

b(i, j)ei,N (nN(q2; q3))e j (s(q1))))

× exp(

2π i(∑

i∈N

b(i, 0)ei,N (nN(q2; q3))e0(q1)))


where ei,N (nN(q2; q3)) is given by (5-7). Also we compute

dcb(q2; q3)= λb(nN(q2; q3); z0)= exp(

2π i(νb(nN(q2; q3))

T

))= exp

(2π i

(∑i∈N

b(i, 0)ei,N (nN(q2; q3)))),

νb(g)= πT

(T∑i∈N

b(i, 0)ei,N (g))∈ R/T Z for g ∈ N ,

with πT : s ∈ R 7→ sT = s+ T Z ∈ R/T Z the quotient map.The last assertion, (5-9), on Hout

m,s(G, N ,T) follows almost automatically fromthe above computations and Lemma 4.6 in the last section.

(iii) We now compute the map

∂πm : Houtm,s(G, N ,T)→ H3(G,T).

We continue to work on the cocycle (λa,b, 1) for a ∈ Za whose restriction to{Hm, K } gives rise to the crossed extension U ∈ Xext(Hm, K ,T), given by

1→ T // Uj←−s j

// K → 1,

where the group K is given by

K = Ker(νb ◦ πG)={g ∈ L :

∑i∈N b( j, 0)e j,N (g) ∈ Z

}.

Then the third cocycle cG ∈ Z3(G,T),

cG(g1; g2; g3)= αsH (g1)(s j (nM(g2; g3)))s j (nM(g1; g2g3))

×(s j (nM(g1; g2))s j (nM(g1g2; g3))

)−1

= λa,b(nM(g2; g3); g1)= λa(nM(g2; g3); g1)

= exp(

2π i( ∑

i< j<k

a(i, j, k)ei (g1)e j (g2)ek(g3)))

= cGa (g1; g2; g3) for g1, g2, g3 ∈ G,

is precisely the image ∂πm◦ δ(λa,b, 1). ♥

6. Concluding remark

The history of cocycle (respectively outer) conjugacy analysis of group actions andgroup outer actions on an AFD factor goes back to the seminal work of Connes[1977; 1976b]. Steady progress was then made over the course of three decades;see especially the work of V. F. R. Jones [1980] and A. Ocneanu [1985].


We have now computed the invariants, which determine the outer conjugacyclass, of an outer action of a countable discrete abelian group on an AFD factorof type IIIλ for 0 < λ < 1. The reduction of outer conjugacy analysis of an outeraction of a countable discrete amenable group on an AFD factor of type IIIλ downto the associated complete invariants was successfully carried out in [Katayamaand Takesaki 2003; 2004; 2007]. As we have shown here, the invariants can becomputed as soon as the group is specified, except in the case of type III0.

Toward the one parameter automorphism group. After completing the classifi-cation of cocycle (respectively outer) conjugacy of countable discrete amenablegroup (respectively outer) actions on an AFD factor, it is natural to consider thesame problem for a continuous group. The first step is obviously to study the one-parameter automorphism group {αt : t ∈R} of an approximately finite-dimensionalfactor R0 of type II1. Indeed, Y. Kawahigashi [1989; 1990; 1991b; 1991a] hasalready classified, up to cocycle (or stable) conjugacy, most one parameter auto-morphism groups of R0 constructed from concrete data; this was extended to thecase of type III by U. K. Hui [2002]. However the general ones with full Connesspectrum are left untouched. One of difficulties is the lack of a technique that wouldallow us to create a one cocycle {us : s ∈R} for a projection p ∈ Proj(R0) such thatthe perturbed one-parameter automorphism group {Ad(ut) ◦ αt : t ∈ R} leaves theprojection p invariant; this would allow us to localize analysis of the action. If aprojection p∈Proj(R0) is differentiable relative to α, then the associated derivationδα generates a desired cocycle. But we don’t know the answer to this:

Question. Does the C∗-algebra

A = {x ∈ R0 : limt→0‖x −αt(x)‖ = 0}

contain a nontrivial projection?

If p∈Proj(A), then for each smooth function f ∈C∞c (R)with compact support,the element

p( f )= α f (p)=∫

R

f (t)αt(p)dt

is smooth, and one can choose f so that ‖p− p( f )‖ is arbitrarily small, so thatSp(p( f )) is concentrated on a neighborhood of the two points {0, 1}; this allows usto generate a nontrivial differentiable projection q near p via the contour integral

q = 12π i

∮|z−1|=r

(z− p( f ))−1dz.

On the other hand, thanks to the exponential functional calculus, one can gener-ate plenty of differentiable unitaries. For example, if h ∈ As.a, then for a real-valuedsmooth function f , we get a differentiable unitary element exp(i f (h)) of A that can


stay near the unitary exp(ih) in norm. Hence the group of differentiable unitariesis σ ∗-strongly dense in the unitary group U(R0).

Acknowledgments

This work originated from the authors’ visit to the Erwin Schrodinger Institute inVienna and the University of Rome in La Sapienza in the spring of 2005 and furtherdeveloped in the subsequent years. Takesaki again visited the Erwin SchrodingerInstitute in fall 2008, where the final touches on the joint work were made. Theauthors are greatly indebted to these institutes, in particular to Professors KlausSchmidt and Sergio Doplicher who made our collaboration possible and pleasant.We record here our sincere appreciation to their support and hospitality.

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Received June 4, 2009.

YOSHIKAZU KATAYAMA

DIVISION OF MATHEMATICAL SCIENCES

OSAKA KYOIKU UNIVERSITY

ASAHIGAOKA 4-698-1KASHIHARA, OSAKA 582-8582JAPAN

[email protected]

MASAMICHI TAKESAKI

DEPARTMENT OF MATHEMATICS

UNIVERSITY OF CALIFORNIA, LOS ANGELES

PO BOX 951555LOS ANGELES, CA 90095-1555UNITED STATES

and

3-10-39 NANKOHDAI

IZUMI-KU, SENDAI 981-8003JAPAN

[email protected]








http://projecteuclid.org/getRecord?id=euclid.pjm/1102650391
















NONHOMOGENEOUS BOUNDARY VALUE PROBLEMSFOR STATIONARY NAVIER–STOKES EQUATIONS

IN A MULTIPLY CONNECTED BOUNDED DOMAIN

HIDEO KOZONO AND TAKU YANAGISAWA

We consider the stationary Navier–Stokes equations on a multiply connectedbounded domain � in Rn for n = 2, 3 under nonhomogeneous boundaryconditions. We present a new sufficient condition for the existence of weaksolutions. This condition is a variational estimate described in terms ofthe harmonic part of solenoidal extensions of the given boundary data; weprove it by using the Helmholtz–Weyl decomposition of vector fields over� satisfying adequate boundary conditions. We also study the validity ofLeray’s inequality for various assumptions about the symmetry of �.

1. Introduction and summary

We consider the stationary Navier–Stokes equations on a bounded domain� in Rn

for n = 2, 3 under nonhomogeneous boundary conditions:

(1-1)

−µ1v+ (v · ∇)v+∇ p = f in �,

div v = 0 in �,

v = β on ∂�.

Here v= v(x)= (v1(x), . . . , vn(x)) and p= p(x) denote the velocity and pressureat x= (x1, . . . , xn)∈�, while f = f (x) and β=β(x)= (β1(x), . . . , βn(x)) denotethe given external force defined on � and the given boundary data defined on ∂�;the coefficient of viscosity is µ > 0. We use standard notation for Laplacian,gradient, divergence, and convective derivative:

1v =∑n

j=1

∂2v

∂x j2 , ∇ p =

(∂p∂x1

, . . . ,∂p∂xn

),

div v =∑n

j=1

∂v j

∂x j, (v · ∇)v =

∑n

j=1v j∂v∂x j

.

MSC2000: 35Q30.Keywords: stationary Navier–Stokes equations, nonhomogeneous boundary value problems,

Helmholtz–Weyl decomposition.

127



128 HIDEO KOZONO AND TAKU YANAGISAWA

Throughout, we use conventional notation such as H m(�), H m0 (�), H s(∂�),

W s,r (�) for m ∈ N, s > 0 and 1 ≤ r ≤ ∞ to denote the usual Sobolev spacesfor either scalar or vector functions. We denote by H 1

0,σ (�) the completion ofC∞0,σ (�) with respect to the Dirichlet norm ‖∇ · ‖L2(�), where C∞0,σ (�) is the set ofu ∈ C∞0 (�) for which div u = 0 in �; we define H 1

0,σ (�)∗ to be the dual space of

H 10,σ (�), and 〈 · , · 〉 denotes the duality pairing between H 1

0,σ (�)∗ and H 1

0,σ (�);the inner product and the norm in L2(�) are denoted by ( · , · ) and ‖·‖, respectively.

We also impose throughout the following assumption on �.

Assumption \. (i) The boundary ∂� has connected components 00, 01, . . . , 0L ,which are C∞ surfaces. The 01, . . . , 0L lie inside 00, and 0i ∩ 0 j = ∅ fori 6= j .

(ii) There exist C∞ surfaces 61, . . . , 6N transverse to ∂� such that 6i ∩6 j =∅for i 6= j , and such that � = � \ 6 is a simply connected domain, where6 =

⋃Nj=16 j .

In the n = 2 case, condition (ii) is always fulfilled and the numbers L in (i)and N in (ii) are equal.

As a consequence of the incompressibility condition div v = 0 of (1-1), theboundary data β is required to satisfy the general flux condition

(GF)L∑

j=0

∫0 j

β · ν d S = 0,

where ν is the outward unit normal to ∂�.Suppose that β ∈ H 1/2(∂�) and f ∈ H 1

0,σ (�)∗. We call v a weak solution of

(1-1) if v ∈ H 1(�) satisfies div v= 0 in �, v= β on ∂�, and the integral identity

(1-2) µ(∇v,∇φ)+ ((v · ∇)v, φ)= 〈 f, φ〉

for all φ ∈ H 10,σ (�). In this paper, we study the existence of weak solutions of

(1-1) under the condition (GF).In his celebrated paper [1933], Leray showed that (1-1) has at least one weak

solution under the restricted flux condition

(RF)∫0 j

β · νd S = 0 for all j = 0, 1, . . . , L ,

which is clearly stronger than the general flux condition (GF). Several fundamentalresults on the existence and regularity of solutions of (1-1) have since been shownby Hopf [1957], Fujita [1961] and Ladyzhenskaya [1969] under the restricted fluxcondition (RF). However, it is still unknown whether there exist solutions of (1-1)with boundary data β satisfying only the general flux condition (GF).

BOUNDARY VALUE PROBLEMS FOR STATIONARY NAVIER–STOKES EQUATIONS 129

One of our main purposes is prove the existence of at least one weak solutionunder a condition weaker than the restricted flux condition (RF). Our sufficientcondition takes the form of a variational estimate (see (1-8) of Theorem 1.3 below)and reflects the topological properties of the domain� explicitly through the spaceVhar(�) of harmonic vector fields over �, defined as the set of h ∈ C∞(�) suchthat div h = 0 and rot h = 0 in �, and h× ν = 0 on ∂�. The boundary conditionappearing in Vhar(�) is different from that usually used in the study of Navier–Stokes equations; see for example [Temam 1979, Theorem 1.5].

In fact, by the Helmholtz–Weyl decomposition of Vhar(�)— see Theorem 2.1 —we can show a useful criterion on solenoidal extensions of the boundary data β:

Proposition 1.1. Let � be a bounded domain in Rn for n = 2, 3 satisfying theassumption (\). Suppose that the boundary data β ∈H 1/2(∂�) satisfies the generalflux condition (GF). Then there exists a solenoidal extension b∈ H 1(�) of β into�such that

(1-3) div b = 0 in � and b = β on ∂�.

Also, any solenoidal extension b ∈ H 1(�) satisfying (1-3) is decomposed as

(1-4) b = h+ rot w,

where h ∈ Vhar(�) and w ∈ X2σ (�)∩ H 2(�), and the following hold:

(I) The vector potential w in (1-4) obeys the estimate

(1-5) ‖w‖H2(�) ≤ c‖β‖H1/2(∂�),

where c is a constant depending only on �,

(II) the harmonic part h in (1-4) is given explicitly as

(1-6) h =L∑`=1

ψ`

L∑j=1

α j`

L∑k=1

α jk

∫0k

β · ν d S.

Here {ψ1, . . . ψL} is the basis of Vhar(�) given below by Theorem 2.1(I) andis related to q j by ψ j =∇q j for j = 1, . . . , L , while (α jk)1≤ j,k≤L is the L×Lregular matrix defined by

(1-7) α jk =

{(1/√1 j−11 j )e jk if 1≤ k ≤ j,

0 if j + 1≤ k ≤ L ,

where e11 = 1 and e jk with 1≤ k ≤ j and j ≥ 2 denotes the ( j, k)-cofactor ofthe matrix

C j =

c11 . . . c1 j.... . .

...

c j1 . . . c j j

for 1≤ j ≤ L


with

c jk =

∫0 j

∂qk

∂νd S = (ψ j , ψk) for j, k = 1, . . . , L ,

and10 = 1 and 1 j = det C j for 1≤ j ≤ L .

The space X2σ (�) appearing in (1-4) is the set of w ∈W 1,2(�) such that divw= 0

in � and w · ν = 0 on ∂�.

Remark 1.2. In view of (1-6), the harmonic part h of b depends only on the basis{ψ j }1≤ j≤L of Vhar(�) and the boundary integrals

∫0 jβ · ν d S for j = 1, . . . , L .

Hence the harmonic part h is independent of the choice of the solenoidal exten-sions b of the boundary data β. Also, h can be regarded as the projection of b ontothe relative de Rham cohomology Vhar(�) of �; see [Schwarz 1995, Section 2.6].

With the aid of Proposition 1.1, we can show our main theorem.

Theorem 1.3. For n = 2, 3, suppose � is a bounded domain in Rn satisfying theassumption (\). Suppose that the boundary data β ∈H 1/2(∂�) satisfies the generalflux condition (GF), and the external force f is in H 1

0,σ (�)∗. Let h be the harmonic

part of the solenoidal extension of β into � given by (1-6).Then, if the estimate

(1-8) supz∈χ(�),∇z 6=0

(h, (z · ∇)z)‖∇z‖2

< µ

holds, there exists at least one weak solution v ∈ H 1(�) of (1-1). Here

(1-9) χ(�)= {z ∈ H 10,σ (�) | ((z · ∇)z, ϕ)= 0 for all ϕ ∈ H 1

0,σ (�)}.

Remark 1.4. (1) The space χ(�) consists of weak solutions of the stationaryEuler equations with Dirichlet boundary condition (see Lemma 2.4). Sucha relation between the existence of weak solutions of (1-1) and the spaceχ(�) above has been already used tacitly in [Leray 1933], [Amick 1984] and[Kapitanskiı and Piletskas 1983] .

(2) By using (1-6), it is not difficult to show that the restricted flux condition (RF)is equivalent to the condition that h≡0 in�. Hence, the existence of solutionsof (1-1) under (RF), already proved in [Leray 1933; Hopf 1957; Fujita 1961;Ladyzhenskaya 1969], can also be derived by applying Theorem 1.3.

As an immediate consequence of Theorem 1.3, we can show that if the harmonicpart h of the solenoidal extension of the boundary data β is small compared to theviscosity µ, then there exist weak solutions of (1-1).

Corollary 1.5. Let �, f , β, and h be as in Theorem 1.3.


(I) Let n = 3. If

(1-10) Cs‖h‖L3(�) < µ,

then there is a weak solution v ∈ H 1(�) of (1-1). Here Cs = 3−1/222/3π−2/3

is the best constant of the Sobolev embedding H 1(�) ↪→ L6(�).

(II) Let n = 2 and let 2< p <∞. If

(1-11) Cq3−1/q1 ‖h‖L p(�) < µ

holds for q = 2p/(p− 2), then there is a weak solution v ∈ H 1(�) of (1-1).Here Cq is the best constant of the Gagliardo–Nirenberg inequality

(1-12) ‖u‖Lq (�) ≤ Cq‖u‖2/qL2(�)‖∇u‖1−2/q

L2(�)for all u ∈ H 1(�) and 2< q <∞,

and31 is the first eigenvalue of the minus Laplace operator−1 under Dirich-let boundary conditions.

Remark 1.6. (1) Galdi [1994, Theorem VIII.4.1] showed in the n = 3 case thatweak solutions of (1-1) exist under a condition somewhat stronger than (1-10).Namely, Galdi assumed that

L∑j=1

k j

∣∣∣∫0 j

β · νd S∣∣∣< ν,

where k j for j = 1, . . . , L , are certain computable constants depending onlyon the domain �. See also [Borchers and Pileckas 1994, Section 1].

(2) In [Kozono and Yanagisawa 2009b], we proved the result stated in (I) usingHopf’s [1957] cut-off function technique and Proposition 1.1. However, theresult in (II) for the n = 2 case seems to be new.

Aside from the corollary above, the variational estimate (1-8) in Theorem 1.3will give us deeper insight into the existence of weak solutions of (1-1). Indeed,by using the variational estimate (1-8), we can systematically study the validity ofLeray’s inequality, whose definition we now recall; see [Takeshita 1993]. Supposethat the boundary data β ∈ H 1/2(∂�) satisfies the general flux condition (GF). Wesay Leray’s inequality (LI) holds for β (and �) if, for an arbitrary ε > 0, there isa solenoidal extension bε of β into � such that div bε = 0 in � and bε = β on ∂�and such that

(LI) |(u · ∇)bε, u)| ≤ ε‖∇u‖2

for all u ∈ H 10,σ (�).


The validity of Leray’s inequality leads to an a priori bound of the Dirichletnorm for all possible weak solutions of (1-1), from which the existence of weaksolutions of (1-1) immediately follows.

In Section 3, we first observe, by using the Helmholtz–Weyl decompositionagain, that if Leray’s inequality holds, the numerator (h, (z ·∇)z) appearing in thevariational estimate (1-8) always vanishes for all z ∈ χ(�); see Proposition 3.1. Inview of this observation, we review the results by Takeshita [1993], Amick [1984]and Fujita [1998], and then give new results on the validity of Leray’s inequalityunder several assumptions about the symmetry of the domain �,

This paper is organized as follows. In Section 2, we recall the Helmholtz–Weyldecomposition of Vhar(�), which we then use to prove Proposition 1.1. Next, weprove Theorem 1.3 with the aid of Leray and Schauder’s fixed point theorem viareduction to absurdity; a key ingredient is Proposition 2.2, a simple observationabout the space χ(�) derived from Proposition 1.1. We then prove Corollary 1.5by using Theorem 1.3. In Section 3, we study the validity of Leray’s inequality.In the appendix, we outline for completeness the proof of the Helmholtz–Weyldecomposition of vector fields over a two-dimensional bounded domain, since weproved it only for the three-dimensional case in [Kozono and Yanagisawa 2009b].

2. Proof of Proposition 1.1, Theorem 1.3, and Corollary 1.5

We first give the Helmholtz–Weyl decomposition of the harmonic space Vhar(�).

Theorem 2.1. Suppose � is a bounded domain in Rn for n = 2, 3 that satisfiesassumption (\).

(I) The space Vhar(�) of harmonic vector fields is L-dimensional. A basis ofVhar(�) is the set {ψ1, . . . , ψL} such that ψ j = ∇q j for j = 1, . . . , L whereq j solves the Dirichlet boundary value problem of the Laplace equation:

(2-1)

{1q j = 0 in �,

q j |0i = δ j i for i = 0, 1, . . . , L .

(II) Let 1< r <∞. For every u ∈ Lr (�), there exist an h ∈ Vhar(�), a w ∈ X rσ (�)

and a p ∈W 1,r0 (�) such that u is decomposed as

(2-2) u = h+ rot w+∇ p in �,

and the triplet {h, w, p} in (2-2) satisfies the estimate

(2-3) ‖h‖Lr (�)+‖w‖W 1,r (�)+‖∇ p‖Lr (�) ≤ C‖u‖Lr (�),

where C is a constant depending only on � and r. This decomposition isunique in that if u = h + rot w + ∇ p with h ∈ Vhar(�), w ∈ X r

σ (�) andp ∈W 1,r

0 (�), then h = h, rot w = rot w and ∇ p =∇ p.


(III) Let 1 < r <∞ and s ≥ 1. If u ∈ W s,r (�) for 1 < r <∞, then the w and pappearing in (2-2) satisfy

w ∈ X rσ (�)∩W s+1,r (�) and p ∈W 1,r

0 (�)∩W s+1,r (�),

and the triplet {h, w, p} in (2-2) satisfies

(2-4) ‖h‖W s,r (�)+‖w‖W s+1,r (�)+‖∇ p‖W s,r (�) ≤ C‖u‖W s,r (�),

where C is a constant depending only on �, s and r.

The space X rσ (�) appearing in statements (II) and (III) is the set ofw∈W 1,r (�)

such that divw = 0 in � and w · ν = 0 on ∂�. When n = 2, rot w in (2-2) shouldbe read as rot w = (∂w/∂x2,−∂w/∂x1) for a scalar function w, and the spacesVhar(�) and X r

σ (�) should be replaced by Vhar(�) and W 1,r0 (�), respectively,

where Vhar(�) is the set of h ∈ C∞(�) such that div h = 0, Rot h = 0 in �and h ∧ ν = 0 on ∂�, with Rot h = ∂h2/∂x1− ∂h1/∂x2 and h ∧ ν = h2ν1− h1ν2.

Proof of Theorem 2.1. In n = 3 case, parts (I) and (II) were proved as [Kozonoand Yanagisawa 2009c, Theorem 1 part (3) and Theorem 3 part (2)], and see also[Bendali et al. 1985]. To prove part (III), we observe that the scalar potential pand the vector potential w in (2-2) are the solutions of two elliptic boundary valueproblems

(2-5)

{1p = div u in �,

p = 0 on ∂�,

and

(2-6)

rot rot w = rot u in �,

divw = 0 in �,

rot w× ν = u× ν on ∂�,

w · ν = 0 on ∂�.

In addition, since −1= rot rot − grad div, we find that (2-6) implies that

(2-7)

−1w = rot u in �,

rot w× ν = u× ν on ∂�,

w · ν = 0 on ∂�.

This casts (2-6) into the form of an elliptic boundary value system with comple-menting boundary conditions in the sense of Agmon, Douglis and Nirenberg; see[Kozono and Yanagisawa 2009c, Lemma 4.3(2)]. Hence, part (III) follows byapplying the regularity theorem of [Agmon et al. 1964] to the boundary value


problems (2-5) and (2-7). The proof of Theorem 2.1 in case when n = 2 will beseparately given in a more general setting in the appendix. �

By using Theorem 2.1, we can prove Proposition 1.1.

Proof of Proposition 1.1. Step 1. Since β ∈ H 1/2(∂�) satisfies (GF), it is wellknown that there exists a solenoidal extension b ∈ H 1(�) of β into � satisfying(1-3) and

(2-8) ‖b‖W 1,2(�) ≤ c‖β‖H1/2(∂�),

where c is a constant depending only on �; see for example [Ladyzhenskaya andSolonnikov 1978].

Step 2. For the solenoidal extension b obtained in the preceding step, we applyTheorem 2.1 with r = 2 to obtain b= h+ rot w+∇ p, where w ∈ X2

σ (�)∩H 2(�),h ∈ Vhar(�), and p ∈W 1,2

0 (�). However, since

1p = div h+ div(rot w)+ div(∇ p)= div b = 0 in �,

and p= 0 on ∂�, we can conclude that p= 0 in �. Therefore, b= h+ rot w. Theestimate (1-5) follows from the estimates (2-4) with s = 1, r = 2 of Theorem 2.1and (2-8).

Step 3. By orthogonalization of the basis {ψ j }Lj=1 of Vhar(�) from Theorem 2.1(I),

we obtain an orthonormal basis

(2-9) ϕ j (x)=L∑

k=1

α jkψk(x) for j = 1, . . . , L ,

where the α jk are the same constants as in (1-7).By virtue of Theorem 2.1(I), we then see from (2-9) that the harmonic part h of

the solenoidal extension b is given as

(2-10)

h =L∑

j=1

(b, ϕ j )ϕ j =

L∑j=1

(b,

L∑k=1

α jkψk

)ϕ j =

L∑j,k=1

α jk(b,∇qk)ϕ j

=−

L∑j,k=1

α jk(div b, qk)ϕ j +

L∑j,k=1

α jkϕ j

∫∂�

(β · ν)qk d S

=

L∑j,k=1

α jkϕ j

∫0k

β · ν d S.


Furthermore, referring to (2-9) again, we have

(2-11)

h =L∑

j,k=1

α jkϕ j

∫0k

β · ν d S.

=

L∑j,k=1

α jk

L∑`=1

α j`ψ`

∫0k

β · ν d S

=

L∑`=1

ψ`

L∑j=1

α j`

L∑k=1

α jk

∫0k

β · ν d S.

This proves (1-6) and thereby Proposition 1.1. �

Proof of Theorem 1.3. The following proposition is crucial for proving Theorem 1.3and is also part of Section 3’s investigation of Leray’s inequality.

Proposition 2.2. Suppose that β ∈ H 1/2(∂�) satisfies (GF). Let b ∈ H 1(�) bean arbitrary solenoidal extension of β into � satisfying (1-3), and let h be theharmonic part of b. Then

(2-12) (b, (z · ∇)z)= (h, (z · ∇)z) for all z ∈ χ(�).

We postpone the proof of Proposition 2.2 to the end of this section. Let b be thesolenoidal extension of β given by Proposition 1.1. Taking u= v−b, we are goingto seek a weak solution u ∈ H 1

0,σ (�) that satisfies

(2-13) µ(∇u,∇φ)+ ((b · ∇)u+ (u · ∇)b+ (u · ∇)u, φ)= 〈F, φ〉

for all φ ∈ H 10,σ (�), where F =µ1b−(b ·∇)b+ f . For this purpose, we introduce

a parameter λ ∈ [0, 1/µ] and the equation

(2-14) (∇uλ,∇φ)+ λ((b · ∇)uλ+ (uλ · ∇)b+ (uλ · ∇)uλ, φ)= 1µ〈F, φ〉,

and put

(2-15) S(λ)= {uλ ∈ H 10,σ (�) | u

λsatisfies (2-14) for all φ ∈ H 10,σ (�)}.

If we can uniformly bound the Dirichlet norm of all uλ ∈ S(λ) as in Lemma 2.3,then (see for example [Ladyzhenskaya 1969] and [Kapitanskiı and Piletskas 1983])the existence of existence of weak solutions u ∈ H 1

0,σ (�) satisfying (2-13) for allφ ∈ H 1

0,σ (�) will easily follow from Leray and Schauder’s fixed point theorem andthe homotopy invariance of the degree of the Leray–Schauder mapping.

Lemma 2.3. If the estimate (1-8) in Theorem 1.3 holds, there exists a constant Msuch that ‖∇uλ‖ ≤ M for all uλ ∈ S(λ), and for all λ ∈ [0, 1/µ].


Proof of Lemma 2.3. We proceed by reduction to absurdity. Suppose that there existsequences {u j }

∞

j=1 ⊂ H 10,σ (�) and {λ j }

∞

j=1 ⊂ [0, 1/µ] satisfying ‖∇u j‖→∞ andλ j → λ0 ∈ [0, 1/µ] as j→∞, and

(2-16) (∇u j ,∇φ)+ λ j ((b · ∇)u j + (u j · ∇)b+ (u j · ∇)u j , φ)=1µ〈F, φ〉

for all φ ∈ H 10,σ (�). Setting φ = u j in (2-16), we have by integration by parts

‖∇u j‖2+ λ j ((u j · ∇)b, u j )=

1µ〈F, u j 〉,

because b satisfies (1-3) and because u j ∈ H 10,σ (�). We then put w j = u j/N j with

N j = ‖∇u j‖ to obtain

(2-17) 1+ λ j ((w j · ∇)b, w j )=1µN j〈F, w j 〉.

Furthermore, since ‖∇w j‖ = 1, we see that the limit w ∈ H 10,σ (�) of {w j }

∞

j=1exists in the sense that

(2-18) ∇w j →∇w weakly in L2(�) and w j → w strongly in L4(�),

as j→∞. Therefore, letting j→∞ in (2-17), we find by (2-18) that

(2-19) 1+ λ0((w · ∇)b, w)= 0.

On the other hand, multiplying both sides of (2-16) by N−2j gives

1N j(∇w j ,∇φ)+

λ j

N j((b ·∇)w j+(w j ·∇)b, φ)+λ j ((w j ·∇)w j , φ)=

1µN 2

j〈F, φ〉,

for all φ ∈ H 10,σ (�). Letting j → ∞ in the above, we can also deduce from

(2-18) that λ0((w · ∇)w, φ) = 0. Since we find by (2-19) that λ0 6= 0, we have((w · ∇)w, φ)= 0 for all φ ∈ H 1

0,σ (�), which implies that w ∈ χ(�).Consequently, if (1-8) in Theorem 1.3 holds, then by Proposition 2.2, we can

see from (2-19) that

1= λ0(b, (w · ∇)w)= λ0(h, (w · ∇)w) < µλ0‖∇w‖2≤ ‖∇w‖2,

which contradicts ‖∇w‖≤1. This proves Lemma 2.3 and thereby Theorem 1.3. �

Proof of Corollary 1.5. In case when n= 3, by Holder’s inequality and the Sobolevembedding theorem we have

(h, (z · ∇)z)≤ ‖h‖L3(�)‖z‖L6(�)‖∇z‖L2(�) ≤ Cs‖h‖L3(�)‖∇z‖2L2(�),

for every z ∈ H 10,σ (�), where Cs is the best constant of the Sobolev embedding

H 1(�) ↪→ L6(�). Therefore, if the condition (1-10) is fulfilled, we see from thisinequality that the estimate (1-8) holds.


In case when n = 2, we combine the inequalities of Holder, Poincare, andGagliardo and Nirenberg to show that

(2-20)

(h, (z · ∇)z)≤ ‖h‖L p(�)‖z‖Lq (�)‖∇z‖L2(�)

≤ Cq‖h‖L p(�)‖z‖2/qL2(�)‖∇z‖2−2/q

L2(�)

≤ Cq3−1/q1 ‖h‖L p(�)‖∇z‖2L2(�),

for all 2 < p < ∞ and 1/q = 1/2− 1/p, where Cq is the best constant of theGagliardo–Nirenberg inequality (1-12) and 31 is the first eigenvalue of −1 underDirichlet boundary conditions. Note that 2< q <∞ since 2< p<∞. Therefore,if (1-11) is satisfied, the estimate (1-8) readily follows from (2-20). �

It remains to prove Proposition 2.2. The following lemma regarding the spaceχ(�) is a slight modification of the result previously proved by Ladyzhenskaya,Kapitanskiı and Piletskas; see also [Amick 1984].

Lemma 2.4 [Kapitanskiı and Piletskas 1983]. For any z ∈ χ(�), there exists ascalar function q ∈W 1,3/2(�) satisfying

(2-21) (z · ∇)z+∇q = 0 in �.

Furthermore, the trace γ (q) ∈W 1/3,3/2(∂�) satisfies

(2-22) γ (q)|0 j = c j for j = 0, 1, . . . , L ,

where c j is a constant that may depend on j .

Proof of Lemma 2.4. Since z ∈ H 1(�), by Holder’s inequality and the Sobolevembedding theorem we see that (z · ∇)z ∈ L3/2(�). Since ((z · ∇)z, ϕ)= 0 for allϕ ∈ H 1

0,σ (�), by applying the Helmholtz decomposition for L3/2(�), we can seethat there exists a scalar function q ∈ W 1,3/2(�) satisfying (z · ∇)z = −∇q in �.That the trace γ (q) takes the constant value c j on each boundary component 0 j

for j = 0, 1, . . . , L is proved in [Kapitanskiı and Piletskas 1983, Lemma 4]. �

Proof of Proposition 2.2. Since the boundary data β satisfies (GF), we can see byProposition 1.1 that the solenoidal extension b ∈ H 1(�) of β into � decomposesas b = h + rot w, with h ∈ Vhar(�) and w ∈ X2

σ (�)∩ H 2(�). Therefore, in viewof Lemma 2.4, one has by integration by parts

(b, (z · ∇)z)= (h+ rot w, (z · ∇)z)

= (h, (z · ∇)z)− (rot w,∇q)

= (h, (z · ∇)z)+∫∂�

(ν×∇γ (q)) ·w d S = (h, (z · ∇)z)

for all z ∈χ(�), because γ (q)|0 j = c j for j =0, 1, . . . , L , and ν×∇ is a tangentialdifferentiation on the boundary. �


3. The validity of Leray’s inequality

We begin with a simple but important observation about the relation between thevalidity of Leray’s inequality and the variational estimate (1-8).

Proposition 3.1. Let � be a bounded domain in Rn for n = 2, 3 satisfying theassumption (\). Suppose that the boundary data β ∈H 1/2(∂�) satisfies the generalflux condition (GF). If Leray’s inequality (LI) holds for β, then

(3-1) (h, (z · ∇)z)= 0 for all z ∈ χ(�).

Here h, the harmonic part of an arbitrary solenoidal extension b ∈ H 1(�) of βinto �, is as given in (1-6).

Proof of Proposition 3.1. Assume that (LI) holds for β ∈ H 1/2(∂�) satisfy-ing (GF). Then, as in the proof of Proposition 1.1, it follows from the Helmholtz–Weyl decomposition that any solenoidal extension bε of β into � decomposes asbε=h+rot wε, where h∈Vhar(�),wε ∈ X2

σ (�)∩H 2(�). Referring to Remark 1.2,we find that h is independent of ε. Whereas, by Proposition 2.2 we have

(bε, (z · ∇)z)= (h, (z · ∇)z) for all z ∈ χ(�).

Therefore, since (LI) holds for β, we see that for an arbitrary ε > 0,

(3-2) |(h, (z · ∇)z)| = |(bε, (z · ∇)z)| = |((z · ∇)bε, z)| ≤ ε‖∇z‖2.

Since h is independent of ε, we can conclude from (3-2) that

(h, (z · ∇)z)= 0 for all z ∈ χ(�). �

The validity of (3-1) for all z ∈ χ(�) implies the estimate (1-8) in Theorem 1.3;we are immediately led from Proposition 3.1 and Theorem 1.3 to this:

Corollary 3.2. Let � and β be as in Proposition 3.1. If Leray’s inequality (LI)holds for β, then there exists at least one weak solution v ∈ H 1(�) of (1-1).

Remark 3.3. Let β ∈ H 1/2(∂�) satisfy the restricted flux condition (RF) and letb ∈ H 1(�) be an arbitrary solenoidal extension of β into �. Since (RF) impliesthat the harmonic part h of b vanishes on �, as mentioned in Remark 1.4(2), wecan see in a way similar to the proof of Proposition 3.1 that

b = rot w for some w ∈ X2σ (�)∩ H 2(�).

Therefore, via Hopf’s cut-off function technique [1957], we can conclude thatLeray’s inequality holds for all β ∈ H 1/2(∂�) satisfying (RF).

In view of Remark 3.3, one might ask whether Leray’s inequality (LI) holdsfor all β ∈ H 1/2(∂�) satisfying only the general flux condition (GF). According to


Takeshita [1993], the answer is no. We will give another proof of Takeshita’s resultby using the following corollary, which is just the contrapositive of Proposition 3.1.

Corollary 3.4. Let �, β and h be as in Proposition 3.1. If there exists a vectorfield z0 ∈ χ(�) such that

(3-3) (h, (z0 · ∇)z0) 6= 0,

then Leray’s inequality (LI) does not hold for β.

Following [Takeshita 1993], we consider the case when � is an annulus in R2

given by �= {x ∈ R2| R1 < |x |< R0} with 0< R1 < R2. We put

00 = {x ∈ R2| |x | = R0} and 01 = {x ∈ R2

| |x | = R1}.

Then, from Theorem 2.1(I), one can see that dim Vhar(�) = 1 and the base ofVhar(�) is given by

h = − x2π |x |2

∫01

β · ν d S.

Take z0= f (|x |)eθ , with nontrivial function f (y)∈C∞0 ((R1, R0)) and unit angularvector eθ . Then it is easy to see that z0 is in H 1

0,σ (�) and (z0 · ∇)z0 =−∇q0(|x |)with q0(r)=

∫ rR1

f 2(s)/s ds. Hence, we see that z0 ∈ χ(�). In addition, we have

(h, (z0 · ∇)z0)=−((z0 · ∇)h, z0)

=1

2π

∫01

β · ν d S( ∫

�

|z0|2

|x |2dx −

∫�

(er · z0)2

|x |dx)

=

∫01

β · ν d S∫ R0

R1

f 2(r)r2 dr,

where er = x/|x |. Therefore, if∫01β ·ν d S 6=0, then (h, (z0 ·∇)z0) 6=0. Combining

Corollary 3.4 and Remark 3.3 then gives another proof of Takeshita’s result.

Proposition 3.5 [Takeshita 1993]. Let � be an annulus domain in R2 as above.Suppose β ∈ H 1/2(∂�) satisfies the general flux condition (GF). Then Leray’sinequality (LI) holds for β if and only if β satisfies the restricted flux condition(RF) as ∫

00

β · ν d S =∫01

β · ν d S = 0.

Remark 3.6. Takeshita [1993, Theorem 2] presented a more general statement:Let � be a bounded domain in Rn with n ≥ 2 and smooth boundary 0 =

⋃Lj=1 0 j ,

where the 0 j are the connected components of 0. Assume that for each suchcomponent there exists a diffeomorphism 8 j of Sn−1

× [0, 1] into � such that8 j (Sn−1

×{0})=0 j and8 j (Sn−1×{1}) is a sphere contained in�. Suppose that

β ∈ H 1/2(∂�) satisfies (GF). Then (LI) holds for β if and only if β satisfies (RF).


Recently, Kobayashi [2009] gave an elementary proof for Takeshita’s statementin the two-dimensional case. In [Kozono and Yanagisawa 2009a], we will give ageneralization of Takeshita’s statement in the three-dimensional case.

From Proposition 3.5 we see that constructive proofs relying on (LI) do notshow the existence of weak solutions of (1-1) under the general flux condition(GF) when � is an annulus. This fact, however, does not mean the nonexistenceof weak solutions of (1-1).

In fact, Amick [1984] showed the existence of weak solutions of (1-1) under(GF) for a class of symmetric domains �⊂ R2, which includes annuli.

Definition 3.7 [Amick 1984]. We say �⊂ R2 has type A symmetry if

(i) � is symmetric with respect to the x1-axis;

(ii) the boundary ∂� has L+1 connected components 00, 01, . . . , 0L , which areC∞ surfaces that each intersect the x1-axis; the 01, . . . , 0L lie inside 00; and0i ∩0 j = φ for i 6= j .

A vector field u = (u1, u2) is said to be symmetric (with respect to the x1-axis)if u1 is an even function of x2 and u2 is an odd function of x2.

Theorem 3.8 [Amick 1984]. Suppose � is a bounded domain in R2 with type Asymmetry and smooth boundary. Suppose that the boundary data βS

∈ H 1/2(∂�)

is symmetric and satisfies the general flux condition (GF), and the external forcef S∈ H 1

0,σ (�)∗ is also symmetric. Then there exists at least one symmetric weak

solution vS∈ H 1(�) of (1-1) with β = βS and f = f S .

Amick proved Theorem 3.8 by showing a uniform a priori estimate similar toLemma 2.3, via reduction of absurdity. The following lemma on the symmetricvector fields of χ(�) was crucial. Define χ S(�) to be the space of all symmetriczS∈ H 1

0,σ (�) such that ((zS· ∇)zS, ϕ)= 0 for all φ ∈ H 1

0,σ (�).

Lemma 3.9 [Amick 1984]. Suppose � is the domain from Theorem 3.8. Supposethat zS

∈ χ S(�) and q S∈ W 1,3/2(�) is the scalar function given in Lemma 2.4

satisfying(zS· ∇)zS

+∇q S= 0 in �.

Then the trace γ (q S) obeys

(3-4) γ (q S)|0 j = C for j = 0, 1, . . . , L ,

where C is a constant independent of j .

On the other hand, by retracing the proof of Theorem 1.3, we get the followingvariant of it in the symmetric case.


Theorem 3.10. Let� be a bounded domain in R2 that satisfies assumption (\) andis symmetric with respect to the x1-axis. Suppose that βS

∈ H 1/2(∂�) is symmetricand satisfies (GF), and the external force f S

∈ H 10,σ (�)

∗ is also symmetric.Then, if

(3-5) supzS∈χ S(�),∇zS 6=0

(hS, (zS· ∇)zS)

‖∇zS‖2< µ,

there exists at least one symmetric weak solution vS∈ H 1(�) of (1-1) with β = βS

and f = f S .Here hS is the harmonic part of an arbitrary solenoidal extension of βS into �

defined by (1-6) with β replaced by βS .

Another proof of Theorem 3.8. Let bS∈ H 1(�) be an arbitrary solenoidal extension

of βS into � and let hs be its harmonic part given by (1-6). Using (3-4) andProposition 2.2, one can see by integration by parts that for all zS

∈ χ S(�)

(hS, (zS· ∇)zS)= (bS, (zS

· ∇)zS)

=−(bS,∇q S)

=−

∫∂�

(βS· ν)γ (q S) d S =−C

∫∂�

βS· ν d S = 0,

because βS satisfies (GF). Thus (3-5) holds for all zS∈ χ S(�), and hence we have

Theorem 3.8 just by applying Theorem 3.10. �

From Takeshita’s statement in Remark 3.6, and from the fact that the proofof Theorem 3.10 still relies on reduction to absurdity, one might be tempted toconclude that even when the domain has type A symmetry and all the data issymmetric, it is still hard to give a constructive proof for the existence of weaksolutions of (1-1). However, Fujita [1998] succeeded in giving such a proof byshowing that Leray’s inequality holds if we consider only symmetric test functionsu ∈ H 1

0,σ (�) in (LI).To make the argument clear, we introduce the symmetric Leray inequality (SLI).

Suppose that � is symmetric with respect to the x1-axis, and the boundary dataβS∈ H 1/2(∂�) satisfies (GF) and is symmetric. We say that the symmetric Leray

inequality holds for βS (and �) if, for arbitrary ε > 0, there exists a symmetricsolenoidal extension bS

ε ∈ H 1(�) of βS into � satisfying the inequality

(SLI) |(uS· ∇)bS

ε , uS)| ≤ ε‖∇uS‖

2

for all symmetric uS∈ H 1

0,σ (�).

Theorem 3.11 [Fujita 1998]. Suppose that � and the boundary data βS are as inTheorem 3.8. Then the symmetric Leray inequality (SLI) holds for βS .


On the other hand, it is easy to see that the argument that proved Proposition 3.1yields the following in the symmetric case.

Proposition 3.12. Let � be a bounded domain in R2 that satisfies assumption (\)and that is symmetric with respect to the x1-axis. Suppose that the boundary dataβS∈ H 1/2(∂�) satisfies the general flux condition (GF) and is symmetric.Then, if the symmetric Leray inequality (SLI) holds for βS , we have

(3-6) (hS, (zS· ∇)zS)= 0 for all zS

∈ χ S(�).

Here hS is the harmonic part of an arbitrary solenoidal extension of βS into �.

Therefore, we can prove the Amick’s result in Theorem 3.8 by just combiningTheorem 3.11 with Proposition 3.12 and Theorem 3.10.

Definition 3.13. We say �⊂ R2 has type B symmetry if

(i) � is symmetric with respect to the x1-axis;

(ii) the boundary ∂� has 2M+1 connected components 00, 01, . . . , 02M , whichare C∞ surfaces; the components 01, . . . , 02M lie inside 00 and 0i ∩0 j = φ

for i 6= j ; and

(iii) the components 02 j−1 and 02 j for j = 1, . . . ,M are symmetric to each otherwith respect to the x1-axis.

Under this symmetry, we will show that the symmetric Leray inequality does nothold for general symmetric boundary data βS satisfying (GF), given an additionalgeometric condition involving the basis of Vhar(�) and the space χ(�).

The following criterion is similar to Corollary 3.4, and is the contrapositive ofProposition 3.12.

Corollary 3.14. Let � and βS be as in Proposition 3.12. If there exists a vectorfield zS

0 ∈ χS(�) such that

(3-7) (hS, (zS0 · ∇)z

S0 ) 6= 0,

then the symmetric Leray inequality (SLI) does not hold for βS .

Let� be a bounded domain in R2 with type B symmetry, and let βS∈ H 1/2(∂�)

satisfy (GF) and be symmetric. We consider first the simplest case that M = 1,which means that ∂� consists of connected components 00, 01 and 02; these areC∞ surfaces such that 01 and 02 lie inside of 00, and 01 and 02 are symmetric toeach other with respect to the x1-axis. We wish to show that there exists a vectorfield zS

0 ∈ χS(�) satisfying (3-7) under an additional geometric condition, when

the boundary data βS does not satisfy (RF). So we first study Vhar(�). Let q1 be asolution of (2-1) with j = 1 and L = 2, so that

(3-8) 1q1 = 0 in �, q1|0 j= δ1 j for j = 0, 1, 2,


and define q2 = q2(x1, x2) by q2(x1, x2) = q1(x1,−x2). Since 01 and 02 aresymmetric to each other with respect to the x1-axis, we find that the q2 above is asolution of (2-1) with j = 2 and L = 2. Hence, we can see that a basis {ψ1, ψ2} ofVhar(�) is given by

(3-9) ψ1(x)=∇q1(x), ψ2(x)=∇q2(x)=(∂q1

∂x1,−∂q1

∂x2

)(x1,−x2).

From (1-6), we can see that the harmonic part hS of an arbitrary solenoidal exten-sion of βS into � is described as

hS=

2∑`=1

ψ`

2∑j=1

α j`

2∑k=1

α jk

∫0k

βS· ν d S

=

2∑`=1

∇q`2∑

j=`

α j`

j∑k=1

α jk

∫0k

βS· ν d S

=∇q1

(α2

11

∫01

βS· ν d S+α2

21

∫01

βS· ν d S+α21α22

∫02

βS· ν d S

)+∇q2

(α22α21

∫01

βS· ν d S+α2

22

∫02

βS· ν d S

)= ((α2

11+α222+α21α22)∇q1+ (α22α21+α

222)∇q2)

∫01

β · ν d S.

In the last equality above, we used the fact that∫01βS·ν d S=

∫02βS·ν d S, which

follows from the symmetry of βS and � with respect to the x1-axis. Therefore, itholds for all zS

∈ χ S(�) that

(hS, (zS· ∇)zS)=

((α2

11+α221+α21α22)(∇q1, (zS

· ∇)zS)

+ (α22α21+α222)(∇q2, (zS

· ∇)zS)) ∫

01

β · ν d S.

We put here kSi = (z

S· ∇)zS

i for i = 1, 2. Since zS is symmetric, we find that k1

and k2 are even and odd, respectively, with respect of x2. Hence, by (3-9) and achange of variables, we have

(∇q2, (zS· ∇)zS)=

∫�

(∂q1

∂x1(x1,−x2)kS

1 (x1, x2)−∂q1

∂x2(x1,−x2)kS

2 (x1, x2))

dx

=

∫�

(∂q1

∂x1(x1, x2)kS

1 (x1,−x2)−∂q1

∂x2(x1, x2)kS

2 (x1,−x2))

dx

=

∫�

(∂q1

∂x1(x1, x2)kS

1 (x1, x2)+∂q1

∂x2(x1, x2)kS

2 (x1, x2))

dx

= (∇q1, (zS· ∇)zS).


The last two displayed equation then give

(hS, (zS· ∇)zS)= (α2

11+α221+ 2α21α22+α

222)(∇q1, (zS

· ∇)zS)

∫01

βS· ν d S

= (α211+ (α21+α22)

2)(∇q1, (zS· ∇)zS)

∫01

βS· ν d S.

By definition, α11 6= 0. Therefore, by Corollary 3.14, the result above gives atheorem:

Theorem 3.15. Let� be a bounded domain in R2 with type B symmetry and M=1.Suppose that the boundary data βS

∈ H 1/2(∂�) is symmetric and satisfies thegeneral flux condition (GF) as∫

00

βS· ν d S+

∫01

βS· ν d S+

∫02

βS· ν d S = 0

but does not satisfy the restricted flux condition (RF), which means that at least oneof these three integrals does not vanish. If there exists a vector field zS

0 ∈ χS(�)

such that

(3-10) (∇q1, (zS0 · ∇)z

S0 ) 6= 0,

then the symmetric Leray inequality (SLI) does not hold for βS . Here q1 is theharmonic function defined by (3-8).

Remark 3.16. By integration by parts, the condition (3-10) is rewritten as

(∇q1, (zS0 · ∇)z

S0 )=−((z

S0 · ∇)∇q1, zS

0 )

=−

2∑i, j=1

∫�

∂2q1

∂xi∂x j(zS

0 )i (zS0 ) j dx =−

∫�

Hess(q1)[zS0 ] dx,

where (zS0 ) j denotes the j-th component of zS

0 and Hess(q1)[zS0 ] stands for the

quadratic form of zS0 associated with the Hessian matrix of q1. However, because

of our lack of our knowledge of the space of χ(�), it seems difficult to check thevalidity of (3-10) so far.

We next study bounded domains � ⊂ R2 with type B symmetry and M ≥ 2.Suppose βS

∈ H 1/2(∂�) is symmetric and satisfies the general flux condition (GF).As before, we let q2 j−1 for j = 1, . . . ,M solve the boundary value problem

(3-11)

{1q2 j−1 = 0 in �,

q2 j−1|0i = δ2 j−1,i for i = 0, 1, . . . , 2M,

and we define q2 j by q2 j (x1, x2) = q2 j−1(x1,−x2) for j = 1, . . . ,M . By thereasoning from the M = 1 case, we then see that these q2 j solve the boundary


value problem

(3-12)

{1q2 j = 0 in �,

q2 j |0i = δ2 j,i for i = 0, 1, . . . , 2M.

It follows from Theorem 2.1(I) that the set {ψ1, . . . , ψ2M}, where

ψ2 j−1(x)=∇q2 j−1(x),

ψ2 j (x)=∇q2 j (x)=(∂q2 j−1

∂x1,−∂q2 j−1

∂x2

)(x1,−x2) for j = 1, . . . ,M,

is a basis of Vhar(�). In the same way as on page 143, one can also see that

(∇q2 j−1, (zS· ∇)zS)= (∇q2 j , (zS

· ∇)zS) for j = 1, . . . ,M.

Therefore, by noting the fact that∫02 j−1

βS·ν d S=

∫02 jβS·ν d S for j = 1, . . . ,M ,

we can derive from the above that

(3-13) (hS, (zS· ∇)zS)

=

2M∑i=1

(∇qi , (zS· ∇)zS)

2M∑j=1

α j i

2M∑k=1

α jk

∫0k

βS· ν d S

=

M∑`=1

(∇q2`−1, (zS· ∇)zS)

2M∑j=1

α j,2`−1

2M∑k=1

α jk

∫0k

βS· ν d S

+

M∑`=1

(∇q2`, (zS· ∇)zS)

2M∑j=1

α j,2`

2M∑k=1

α jk

∫0k

βS· ν d S

=

M∑`=1

(∇q2`−1, (zS· ∇)zS)

2M∑j=1

(α j,2`−1+α j,2`)

2M∑k=1

α jk

∫0k

βS· ν d S

=

M∑`=1

(∇q2`−1, (zS· ∇)zS)

×

2M∑j=1

M∑k=1

(α j,2`−1+α j,2`)(α j,2k−1+α j,2k)

∫02k−1

βS· ν d S

Now for `= 1, . . . ,M , we put

(3-14) p`[βS] =

2M∑j=1

M∑k=1

(α j,2`−1+α j,2`)(α j,2k−1+α j,2k)

∫02k−1

βS· ν d S


and q`[zS] = (∇q2`−1, (zS

· ∇)zS). Then we can rewrite (3-13) as

(3-15) (hS, (zS· ∇)zS)= 〈P[βS

], Q[zS]〉RM ,

where

P[βS] = (p1[β

S], . . . , pM [β

S]) and Q[zS

] = (q1[zS], . . . , qM [zS

]),

and 〈 · , · 〉RM denotes the inner product in RM . The next lemma shows that thetriviality of P[βS

] implies the restricted flux condition (RF).

Lemma 3.17. Let� be a bounded domain in R2 with type B symmetry and M ≥ 2.Suppose βS

∈ H 1/2(∂�) is symmetric and satisfies the general flux condition (GF).Then, P[βS

] = 0 if and only if the restricted flux condition (RF) holds as∫0 j

βS· ν d S = 0 for all j = 0, 1, . . . , 2M.

Proof of Lemma 3.17. We first observe from (3-14) that

(3-16)

tP[βS] =

( 2M∑j=1

(α j,2`−1+α j,2`)(α j,2k−1+α j,2k)∣∣∣ ` ↓ 1, . . . ,M, k→ 1, . . . ,M

)

×

(∫02k−1

βS· ν d S

∣∣∣ k ↓ 1, . . . ,M).

On the other hand, a straightforward calculation yields

det( 2M∑

j=1

(α j,2`−1+α j,2`)(α j,2k−1+α j,2k)∣∣∣ ` ↓ 1, . . . ,M, k→ 1, . . . ,M

)

=

∑1≤ j1<···< jM≤2M

( ∑σ=

( j1,..., jMk1,...,kM

) sgn(σ )(αk1,1+αk1,2) · · · (αkM ,2M−1+αkM ,2M)

)2

,

where sgn(σ ) is the sign of the permutation σ . It is easy to see that the right side ofthis equation contains the term (α11α33 · · ·α2M−1,2M−1)

2, which is positive by thedefinition of the α jk in Proposition 1.1. Hence, the determinant above is nonzero.Therefore, it follows from (3-16) that P[βS

] = 0 if and only if∫02 j−1

βS· ν d S = 0 for all j = 1, . . . ,M.

It is easy to see that this is equivalent to (RF) since βS and � are symmetric withrespect to the x1-axis and βS satisfies (GF). This proves Lemma 3.17. �

Accordingly, using Corollary 3.14 again and referring to Lemma 3.17, we havethe following theorem.


Theorem 3.18. Let� be a bounded domain in R2 with type B symmetry and M≥2.Suppose that the boundary data βS

∈ H 1/2(∂�) is symmetric and satisfies thegeneral flux condition (GF) as

2M∑j=0

∫0 j

βS· ν d S = 0,

but does not satisfy the restricted flux condition (RF), which means that at least oneof the integrals in the previous expression does not vanish. If there exists a vectorfields zS

0 ∈ χS(�) such that

(3-17) 〈P[βS], Q[zS

0 ]〉RM 6= 0,

then the symmetric Leray inequality (SLI) does not hold for βS .

Remark 3.19. As can seen from the argument above, it is not difficult to generalizeTheorem 3.18 to the case of Rn with n ≥ 3.

Appendix

We here outline a proof of the Helmholtz–Weyl decomposition of vector fields overtwo-dimensional bounded domains; this decomposition is more general than then = 2 case of Theorem 2.1. In this appendix, we let

Xhar(�)= {h ∈ C∞(�) | div h = 0,Rot h = 0 in �, h · ν = 0 on ∂�},

Vhar(�)= {h ∈ C∞(�) | div h = 0,Rot h = 0 in �, h ∧ ν = 0 on ∂�},

where Rot h = ∂h2/∂x1 − ∂h1/∂x2 and h ∧ ν = h2ν1 − h1ν2 for a vector-valuedfunction h = (h1, h2), and rot w = (∂w/∂x2,−∂w/∂x1) for a scalar function w.

The aim here is to show the following theorem.

Theorem 3.20. Let � be a bounded domain in R2 satisfying the assumption (\).

(I) The spaces Xhar(�) and Vhar(�) are L-dimensional. Furthermore, a basis{ϕ1, . . . , ϕL} of Xhar(�) and a basis {ψ1, . . . , ψL} of Vhar(�) are given by

ϕ j = rot q j and ψ j = grad q j for j = 1, . . . , L ,

respectively, where the q j are solutions of the following Dirichlet boundaryvalue problem for the Laplace equation:

1q j = 0 in � and q j |0i = δi j for i = 0, 1, . . . , L .

(II) Let 1< r <∞. For every u ∈ Lr (�),

(a) there exists an h ∈ Xhar(�), a w ∈W 1,r0 (�) and a p ∈W 1,r (�) such that

u = h+ rot w+∇ p in �, or


(b) there exists an h ∈ Vhar(�), a w ∈ W 1,r (�) and a p ∈ W 1,r0 (�) such that

u = h+ rot w+∇ p in �.

In both cases (a) and (b), the triplet {h, w, p} is subject to the estimate

(3-18) ‖h‖Lr (�)+‖w‖W 1,r (�)+‖∇ p‖Lr (�) ≤ C‖u‖Lr (�),

where C is a constant depending only on � and r. The decompositions in (a)and (b) are unique in the same sense as in Theorem 2.1(II).

(III) Let 1< r <∞ and s ≥ 1. If u ∈ W s,r (�), then the w and p appearing in thedecomposition (a) or (b) gain further regularity such that

w ∈W 1,r0 (�)∩W s+1,r (�) and p ∈W s+1,r (�) in case (a),

or

w ∈W s+1,r (�) and p ∈W 1,r0 (�)∩W s+1,r (�) in case (b).

In both cases (a) and (b), the triplet {h, w, p} is subject to the estimate

(3-19) ‖h‖W s,r (�)+‖w‖W s+1,r (�)+‖∇ p‖W s,r (�) ≤ C‖u‖W s,r (�),

where C is a constant depending only on �, s and r.

Proof. The proof proceeds in almost the same way as in [Kozono and Yanagisawa2009c, Theorem 1]; we call this “our other paper” here. Our other paper studiedonly the case of three-dimensional bounded domains. Here we will point out onlythe differences.

Given a vector field u ∈ Lr (�), it is not difficult to see that the scalar functionsp and w appearing in case (a) or (b) formally satisfy the following boundary valueproblems: In case (a),

1p = div u in �,

∂p∂ν= u · ν on ∂�,

(3-20)

{Rot rot w = Rot u in �,

w = 0 on ∂�,(3-21)

and in case (b) {1p = div u in �,

p = 0 on ∂�,(3-22) Rot rot w = Rot u in �,

∂w∂ν= u ∧ ν on ∂�.

(3-23)


Since the governing boundary value problems (3-20) and (3-22) for p are the sameas those in the three-dimensional case, we need only investigate the governingboundary value problems (3-21) and (3-23) for w. As in our other paper, we arereadily led to weak formulations of solutions of (3-21) and (3-23): In case (a), ascalar function w ∈W 1,r

0 (�) is said to be a weak solution of (3-21) if

(3-24) (rot w, rot ϕ)= (u, rot ϕ)

for any scalar functions ϕ ∈ W 1,r ′0 (�) with r ′ = r/(r − 1); in case (b), a scalar

function w ∈W 1,r (�) is a weak solution of (3-23), if

(3-25) (rot w, rot ϕ)= (u, rot ϕ)

for any scalar functions ϕ ∈W 1,r ′(�) with r ′ = r/(r − 1).Then we can easily see that the same procedure from our other paper still works

to establish the Lr -variational inequalities associated with the weak formulations(3-24) and (3-25). By using those Lr -variational inequalities, we achieve the exis-tence of weak solutions of (3-21) and (3-23). The rest of the proof is word-for-wordrepetition of the proof in our other paper. �

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Received January 23, 2009. Revised May 16, 2009.

HIDEO KOZONO

MATHEMATICAL INSTITUTE

TOHOKU UNIVERSITY

SENDAI 980-8578JAPAN

[email protected]

TAKU YANAGISAWA

DEPARTMENT OF MATHEMATICS

NARA WOMEN’S UNIVERSITY

NARA 630-8506JAPAN

[email protected]



http://dx.doi.org/10.1007/s00209-008-0361-2

http://dx.doi.org/10.1007/s00209-008-0361-2







http://www.emis.de/cgi-bin/JFM-item?59.0402.01

http://www.emis.de/cgi-bin/JFM-item?59.0402.01











HYPERBOLIC STRUCTURES ON CLOSED SPACELIKEMANIFOLDS

KUN ZHANG

We give an intrinsic generalization of spacelike manifolds. We define theintrinsic mean curvature flow and study it on certain closed generalizedspacelike manifolds. Then we prove the existence of hyperbolic structureson them.

1. Introduction

Recall that a Riemannian manifold (M, g) is hyperbolic if it has constant negativesectional curvature. These manifolds all come from the quotient of hyperbolicspace H n by discrete isometry groups. However, it is difficult to find a good in-trinsic characterization for whether hyperbolic structures exist on a given manifold.First, we know that some negatively pinched Riemannian manifolds do not admita hyperbolic metric. The n ≥ 4 counterexample in [Gromov and Thurston 1987]contrasts sharply with the pinching theorem of positively curved manifolds, andimplies that is it not always possible to deform by geometric flow a given negativelycurved metric into one with constant negative curvature. However, this paper willshow that a hyperbolic structure exists naturally on a large class of manifolds.

Consider the well known model of hyperbolic space by the imaginary unit spherein Minkowski space R1,n , where the Minkowski metric in Cartesian coordinates(x0, x1, . . . , xn) is

g =−(dx0)2+ (dx1)2+ · · ·+ (dxn)2

and the equation of the imaginary unit sphere is

−(x0)2+ (x1)2+ · · ·+ (xn)2 =−1.

That the imaginary unit sphere has sectional curvature equal to −1 can be seenfrom the Gauss–Codazzi equations

Ri jkl − (hilh jk − hikh jl)= 0 and ∇i h jk −∇ j hik = 0,

MSC2000: 53C44.Keywords: hyperbolic structures, closed spacelike manifolds, mean curvature flow.

151



152 KUN ZHANG

where hi j is the second fundamental form, and hi j equals gi j on the imaginary unitsphere. In this paper, we are interested in an intrinsic generalization of this model.

Definition 1.1. We call a triple (M, gi j , hi j ) a spacelike manifold if (M, gi j ) is aRiemannian manifold and hi j is a symmetric tensor satisfying the Gauss–Codazziequations

Ri jkl − (hilh jk − hikh jl)= 0 and ∇i h jk −∇ j hik = 0.

Remark 1.2. It follows from this definition that there is a locally isometric andspacelike embedding of (M, g) into R1,n , and we can globally embed the universalcover of (M, g) into R1,n as a spacelike hypersurface.

Our main theorem is this:

Theorem 1.3. Let (M, g, h) be an n-dimensional closed spacelike manifold withhi j > 0 and n ≥ 4. Then M admits a hyperbolic metric.

The idea is to use geometric flows. In contrast with extrinsic mean curvatureflow, we define an intrinsic mean curvature flow of (g, h) by

(1-1)∂t gi j =−2Ri j + 2himhnj gmn,

∂t hi j =4hi j − Rimhnj gmn− R jmhni gmn

+ 2hikhlmhnj gkl gmn− |A|2hi j ,

with gi j (x, 0)= gi j (x), the initial metric on M , and hi j (x, 0)= hi j (x), the initialdata of hi j . Here |A|2 = gik g jlhi j hkl and ∂t := ∂/∂t . We will solve (1-1) intrinsi-cally and show that the solution exists for all times in [0,∞) and converges (afternormalization) to a hyperbolic metric.

Remark 1.4. Mean curvature flows have been intensively studied in recent years.See [Huisken 1984] for Euclidean ambient space and [Ecker and Huisken 1991;Ecker 1997] for Minkowski ambient space. Note that in extrinsic mean curvatureflow (with ambient space R1,n), we deform the position vector F by the evolutionequation ∂F/∂t = −H . With the Gauss–Codazzi equations we begin with theequations of the metric and the second fundamental form that come from the ex-trinsic mean curvature flow, and change them to the weakly parabolic system (1-1).This system is intrinsically defined and interesting in its own right.

Remark 1.5. In [Ecker 1997], it was shown that there is a long-time solution formean curvature flow of noncompact spacelike hypersurfaces in Minkowski space.If we were to lift (M, g) to its universal cover and deform the universal cover bythe extrinsic mean curvature flow, we would get a long-time solution. Then toget an induced solution on M , we would need a uniqueness theorem for the meancurvature flow. Here, we try a completely different method.

HYPERBOLIC STRUCTURES ON CLOSED SPACELIKE MANIFOLDS 153

2. Short-time existence and uniqueness

System (1-1) is not strictly parabolic, so in order to apply the theory of such systemsto get short-time existence, we will use a trick of De Turck: We will combine ourevolution (1-1) with the harmonic map flow.

Let (Mn, gi j (x)) and (N m, sαβ(y)) be Riemannian manifolds with F :Mn→N m

a map between them. The harmonic map flow is an evolution equation for mapsfrom Mn to N m and is given by

(2-1)∂t F(x, t)=4F(x, t) for x ∈ Mn and t > 0,

F(x, 0)= F(x) for x ∈ Mn,

where 4 is defined by using the metrics gi j (x) and sαβ(y) through

4Fα(x, t)= gi j (x)∇i∇ j Fα(x, t),

and

(2-2) ∇i∇ j Fα(x, t)= ∂2 Fα

∂x i∂x j −0ki j∂Fα

∂xk + 0αβγ∂Fβ

∂x i∂Fγ

∂x j .

Here we use {x i} and {yα} to denote the local coordinates of Mn and N m , respec-

tively, and 0ki j and 0αβγ are the corresponding Christoffel symbols of gi j and sαβ .

The harmonic map flow is strictly parabolic, so for any initial data, there exists ashort-time smooth solution.

Let (gi j (x, t), hi j (x, t)) be a complete smooth solution of our system (1-1).Then the harmonic map flow coupled with our evolution equation is the system

(2-3)∂t F(x, t)=4t F(x, t) for x ∈ Mn and t > 0,

F(x, 0)= identity for x ∈ Mn,

where 4t is defined by using the metrics gi j (x, t) and sαβ(y).Let (F−1)∗g and (F−1)∗h be the one-parameter families of pulled-back met-

rics and tensors on the target (N n, sαβ). Write gαβ(y, t) = ((F−1)∗g)αβ(y, t) andhαβ(y, t)= ((F−1)∗h)αβ(y, t). Then by direct calculations, gαβ(y, t) and hαβ(y, t)satisfy the evolution equations

(2-4)

∂t gαβ(y, t)=−2Rαβ(y, t)+ 2hασ hρβ gσρ +∇αVβ +∇βVα,

∂t hαβ(y, t)=4hαβ(y, t)− Rασ hρβ gσρ − Rβσ hρα gσρ

+ 2hαλhµν hρβ gλµgνρ − | A|2hαβ + hβγ∇αV γ+ hαγ∇βV γ ,

where V α= gβγ (0αβγ (g)− 0

αβγ (s)), and 0αβγ (g) and 0αβγ (s) are the Christoffel

symbols of the metrics gαβ(y, t) and sαβ(y), respectively. Here we analyze the

154 KUN ZHANG

principal part of the right side of (2-4). One can see that

−2Rαβ(y, t)+ 2hασ hρβ gσρ +∇αVβ +∇βVα = gµν∂2gαβ∂yµ∂yν

+ (lower order terms)

and

4hαβ(y, t)− Rασ hρβ gσρ − Rβσ hρα gσρ

+ 2hαλhµν hρβ gλµgνρ − | A|2hαβ + hβγ∇αV γ+ hαγ∇βV γ

= gµν(∂2hαβ∂yµ∂yν

−∂0σαµ

∂yνhσβ −

∂0σβµ

∂yνhσα

)− gµν

(−∂0σαµ

∂yν+∂0σµν

∂yα

)hσβ

− gµν(−∂0σβµ

∂yν+∂0σµν

∂yβ

)hσα + gµν

∂0γµν

∂yαhγβ + gµν

∂0γµν

∂yβhγα

+ (lower order terms)

= gµν∂2hαβ∂yµ∂yν

+ (lower order terms).

Hence

(2-5)∂t gαβ(y, t)= gµν

∂2gαβ∂yµ∂yν

+ (lower order terms),

∂t hαβ(y, t)= gµν∂2hαβ∂yµ∂yν

+ (lower order terms),

and we know (2-4) is a strictly parabolic system. By the theory of such equations,there exists a smooth short-time solution of (2-4) for any initial data.

We can recover the solution (g, h) for the original evolution equations from thesolution (g, h), as follows. Let (N n, sαβ)= (Mn, gαβ( · , 0)). Since

(2-6) V α= gβγ (0αβγ (g)− 0

αβγ (s))=−(4F ◦ F−1)α,

we have

(2-7) ∂t F =−V ◦ F.

Now once we have gαβ , we know V and can solve (2-7), which is just a system ofordinary differential equations on the domain M . Hence (g, h) can be recoveredas the pull-backs g = F∗g and h = F∗h.

Now we claim uniqueness of the solutions of (1-1) with given smooth initialconditions on a compact manifold. Suppose (g1, h1) and (g2, h2) are two solutionsthat agree at t = 0. We can solve the coupled harmonic map flow (2-3) for mapsF1 and F2 with the metrics g1 and g2 on M into the same target N , with the samefixed s and initial data. Then we have two solutions g1 and g2 on N with the sameinitial metric. By the standard uniqueness result for strictly parabolic equations, we


have (g1, h1)= (g2, h2). Hence by (2-6) the corresponding vector fields V1 = V2.Then the solutions of the ordinary differential equations ∂t F1 = −V1 ◦ F1 and∂t F2 = −V2 ◦ F2 with the same initial values must coincide, and the solutions(g1, h1)= F∗(g1, h1) and (g2, h2)= F∗(g2, h2) of (1-1) must agree.

3. Preservation of the Gauss–Codazzi equations

Here we will show that the Gauss–Codazzi equations are preserved under (1-1).Let Gi jkl = Ri jkl − (hilh jk − hikh jl) and Ci jk =∇i h jk −∇ j hik .

Proposition 3.1. If the tensor hi j satisfies the Gauss–Codazzi equations

Ri jkl − (hilh jk − hikh jl)= 0 and ∇i h jk −∇ j hik = 0

at time t = 0, then it also does so for t > 0.

Proof. By direct calculations, we have

∂t0ki j =

12 gkl(∇ j (∂t gil)+∇i (∂t g jl)−∇l(∂t gi j )),

∂t Rki jl =∇i (∂t0

kjl)−∇ j (∂t0

kil),

∂t Ri jkl = ghk∂t Rhi jl + ∂t ghk Rh

i jl .

With these identities we get

∂t Ri jkl =∇i∇k R jl −∇i∇l R jk −∇ j∇k Ril +∇ j∇l Rik

−∇i∇k(h jmhnl gmn)+∇i∇l(h jmhnk gmn)

+∇ j∇k(himhnl gmn)−∇ j∇l(himhnk gmn)

− Ri jks(Rtl − htmhnl gmn)gst− Ri jsl(Rtk − htmhnk gmn)gst

and the identity

4Ri jkl=−2(Bi jkl−Bi jlk−Bil jk+Bik jl)+∇i∇k R jl−∇i∇l R jk−∇ j∇k Ril+∇ j∇l Rik

+ Rmjkl Rni gmn+ Rimkl Rnj gmn,

where Bi jkl = Rmi js Rnklt gmngst . Then we obtain

(3-1) (∂t −4)Ri jkl − 2(Bi jkl − Bi jlk − Bil jk + Bik jl)

=−Ri jks(Rtl − htmhnl gmn)gst− Ri jsl(Rtk − htmhnk gmn)gst

− Rs jkl(Rti − htmhni gmn)gst− Riskl(Rt j − htmhnj gmn)gst

− Rs jklhtmhni gmngst− Risklhtmhnj gmngst

−∇i∇k(h jmhnl gmn)+∇i∇l(h jmhnk gmn)

+∇ j∇k(himhnl gmn)−∇ j∇l(himhnk gmn).

156 KUN ZHANG

To simplify the evolution equations, we will use a moving frame trick. Let uspick an abstract vector bundle V over M isomorphic to the tangent bundle T M .Choose an orthonormal frame Fa = F i

a∂/∂x i for a = 1, . . . , n of V at t = 0; thenevolve Fa

i by the equation

∂t F ia = gi j (R jk − h jmhnk gmn)Fk

a .

Then the frame F = {F1, . . . , Fa, . . . , Fn} will remain orthonormal for all time.We will use indices a, b, . . . on a tensor to denote its components in the evolvingorthonormal frame. In this frame we have

(3-2) (∂t −4)Rabcd − 2(Babcd − Babdc− Badcb+ Bacbd)

=−Rsbcdhtmhnagmngst− Rascdhtmhnbgmngst

−∇a∇c(hbmhnd gmn)+∇a∇d(hbmhncgmn)

+∇b∇c(hamhnd gmn)−∇b∇d(hamhncgmn)

and

(3-3) (∂t −4)hab =−|A|2hab.

By calculations, we have

(3-4) (∂t −4)(Rabcd − (hadhbc− hachbd))

= 2(Babcd − Babdc− Badcb+ Bacbd)

− Rsbcdhtmhnagmngst− Rascdhtmhnbgmngst

−∇a∇c(hbmhnd gmn)+∇a∇d(hbmhncgmn)

+∇b∇c(hamhnd gmn)−∇b∇d(hamhncgmn)

+ 2|A|2(hadhbc− hachbd)+ 2(∇mhad∇nhbc−∇mhac∇nhbd)gmn.

Then we want to replace Babcd by

Babcd =(Rmabs − (hmshab− hmbhas)

)(Rmcds − (hmshcd − hmdhcs)

)gmngst

and replace terms including ∇h and ∇∇h by C and ∇C , respectively. That is,

Babcd − Babdc− Badcb+ Bacbd(3-5)

= Babcd − Babdc− Badcb+ Bacbd

− Rmabshdnhtcgmngst− Rmcdshbnhtagmngst

+ Rmabshcnhtd gmngst+ Rmdcshbnhtagmngst

− Rmadshnt hcbgmngst+ Rmadshcnhtbgmngst

− Rmbcshnt had gmngst+ Rmbcshdnhtagmngst


+ Rmacshnt hdbgmngst− Rmacshdnhtbgmngst

+ Rmbdshnt hacgmngst− Rmbdshcnhtagmngst

− hamhbshcnhdt gmngst+ hamhbshdnhct gmngst

+ hadhbc|A|2− hamhdshnt hbcgmngst− hbmhcshnt had gmngst

− hachbd |A|2+ hamhcshnt hbd gmngst+ hbmhdshnt hacgmngst

and

(3-6) −∇a∇c(hbmhnd gmn)+∇a∇d(hbmhncgmn)+∇b∇c(hamhnd gmn)

−∇b∇d(hamhncgmn)+ 2(∇mhad∇nhbcgmn−∇mhac∇nhbd gmn)

=−∇c(∇ahbm −∇bham)hnd gmn−∇a(∇chdm −∇dhcm)hnbgmn

+∇d(∇ahbm −∇bham)hncgmn−∇b(∇chdm −∇dhcm)hnagmn

− (∇ahbm −∇bham)(∇chdn −∇dhcn)gmn

− (∇ahdm −∇mhad)∇chbngmn− (∇dham −∇mhad)∇bhcngmn

+ (∇ahcm −∇mhac)∇dhbngmn+ (∇cham −∇mhac)∇bhdngmn

+ (∇mhbc−∇chmb)∇nhad gmn+ (∇mhbc−∇bhmc)∇nhad gmn

− (∇mhbd −∇dhmb)∇nhacgmn− (∇mhbd −∇bhmd)∇nhacgmn

− Racbmhnshtd gmngst− Racmshndhtbgmngst

+ Rbcamhnshtd gmngst

+ Rbcmshndhtagmngst+ Radbmhnshtcgmngst

+ Radmshnchtbgmngst

− Rbdamhnshtcgmngst− Rbdmshnchtagmngst .

Let us denote the curvature tensor by Rm and denote any tensor product of tensorsS and T by S ∗T when we do not need the precise expression. If we replace termsincluding Rm ∗h ∗ h by terms G ∗ h ∗ h and if we use (3-4), (3-5) and (3-6), thensome calculations gives

(3-7) (∂t −4)G = G ∗G+G ∗ h ∗ h+∇C ∗ h+C ∗∇h+C ∗C,

where Gi jkl = Ri jkl − (hilh jk − hikh jl) and Ci jk =∇i h jk −∇ j hik . Since

∂t∇i h jk =∇i (∂t h jk)− (∂t0li j )hlk − (∂t0

lik)hl j

=∇i (4h jk − R jmhnk gmn− Rkmhnj gmn

+ 2h jmhnshtk gmngst− |A|2h jk)

− (∂t0li j )hlk +∇i Rkmhnj gmn

+∇k Rimhnj gmn−∇m Rikhnj gmn

−∇i hkmhnsht j gmngst−∇i hmshnkht j gmngst

−∇khimhnsht j gmngst

−∇khmshni ht j gmngst+∇mhishnj htk gmngst

+∇mhkshnj hti gmngst

158 KUN ZHANG

and

4(∇i h jk)= gmn∇m∇n(∇i h jk)

=∇i (4h jk)+ Rim∇nh jk gmn+ 2(Rmi js∇nhtk + Rmiks∇nht j )gmngst

+∇ j Rimhnk gmn−∇m Ri j hnk gmn

+∇k Rimhnj gmn−∇m Rikhnj gmn,

we get

(3-8) (∂t −4)∇i h jk + (∂t0li j )hlk

=−R jm∇i hnk gmn− Rkm∇i hnj gmn

− Rim∇nh jk gmn

+∇i (2h jmhnshtk gmngst− |A|2h jk)

−∇i R jmhnk gmn−∇ j Rimhnk gmn

− 2(Rmi js∇nhtk + Rmiks∇nht j )gmngst

−∇i hkmhnsht j gmngst−∇i hmshnkht j gmngst

−∇khimhnsht j gmngst−∇khmshni ht j gmngst

+∇mhishtkhnj gmngst+∇mhkshti hnj gmngst .

Then in the moving frame we obtain

(3-9) (∂t −4)∇ahbc+ |A|2∇ahbc+ (∂t0li j )hlk F i

a F jb Fk

c

=−∇ahcmhnshtbgmngst−∇ahmbhnshtcgmngst

−∇mhbchnshtagmngst

+ 2∇ahbmhnshtkcgmngst+ 2∇ahcmhnshtbgmngst

+ 2∇ahmshnbhtcgmngst− 2∇ahmshnt hbcgmngst

−∇ahcmhnshtbgmngst−∇ahmshnbhtcgmngst

−∇chamhnshtbgmngst

+∇mhashnbhtcgmngst+∇mhcshnbhtagmngst

− 2Rmabs∇nhtcgmngst− 2Rmacs∇nhtbgmngst .

Then we replace terms including ∇h by C and terms including Rm by G. Finally,we have

(3-10) (∂t −4)C =−|A|2C +C ∗ h ∗ h+C ∗Rm+G ∗∇h.

Combing (3-7) and (3-10), we obtain

(3-11)

(∂t −4)(|G|2+ |C |2)

≤ C1(|G|2+ |C |2)− 2|∇G|2− 2|∇C |2

+〈G,G ∗G+G ∗ h ∗ h+∇C ∗ h+C ∗∇h+C ∗C〉

+ 〈C,−|A|2C +C ∗ h ∗ h+C ∗Rm+G ∗∇h〉

≤ C2(|G|2+ |C |2),


where we use the Cauchy–Schwarz inequality, and for 0≤ t < δ we have bounded|Rm|, |A| and |∇h|. Thus, by the standard maximum principle

ddt(|G|2+ |C |2)max ≤ C2(|G|2+ |C |2)max,

we get(|G|2+ |C |2)max(t)≤ eC2t(|G|2+ |C |2)max(0).

Since (|G|2+|C |2)max(0)= 0, the Gauss–Codazzi equations are preserved as longas the solution exists. �

In the following we will still call hi j (x, t) the second fundamental form and itstrace H the mean curvature.

4. Evolution of the metric and curvature

Using the Gauss–Codazzi equations, we rewrite our evolution equations:

Proposition 4.1.

∂t gi j = 2Hhi j .(4-1a)

(∂t −4)hi j = 2Hhimhnj gmn− |A|2hi j .(4-1b)

(∂t −4)H =−H |A|2.(4-1c)

(∂t −4)|A|2 =−2|∇A|2− 2|A|4.(4-1d)

Since hi j is positive at t =0 and M is compact, there are some ε>0 and β >0 suchthat βHgi j ≥ hi j ≥ εHgi j holds on M at t = 0. We want to show that inequalityremains true as long as the solution of our evolution equations (1-1) exists. Forthis purpose we need the following maximum principle for tensors on manifolds,which is proved in [Hamilton 1982].

On a compact manifold M , let uk be a vector field and Mi j and Ni j be symmetrictensors, all of which may depend on time t . Assume that Ni j = p(Mi j , gi j ) is apolynomial in Mi j formed by contracting products of Mi j with itself using themetric. Suppose this polynomial satisfies the condition Ni j X i X j

≥ 0 for any null-eigenvector X of Mi j .

Theorem 4.2 [Hamilton 1986]. Suppose that the evolution equation

∂t Mi j =4Mi j + uk∇k Mi j + Ni j

holds on 0≤ t<T , where Ni j = p(Mi j , gi j ) satisfies the null-eigenvector conditionabove. If Mi j ≥ 0 at t = 0, then it remains so on 0≤ t < T .

Proposition 4.3. If εHgi j ≤ hi j ≤ βHgi j and H > 0 at t = 0, then these relationscontinue to hold as long as the solution of (1-1) exists.

160 KUN ZHANG

Proof. First, by using maximum principle on the equation (∂t −4)H = −H |A|2,we know H > 0 as long as the solution of (1-1) exists. Then we consider

Mi j = hi j − εHgi j ,

∂t Mi j = ∂t hi j − ε∂t Hgi j − εH∂t gi j

=4hi j + 2Hhimhnj gmn− |A|2hi j − ε(4H − |A|2 H)gi j − εH(2Hhi j )

=4Mi j + 2Hhimhnj gmn− |A|2(hi j − εHgi j )− 2εH 2hi j .

For any null vector vi of Mi j , we have

(2Hhimhnj gmn− |A|2(hi j − εHgi j )− 2εH 2hi j )v

j

= 2Hhim gmn(εHvn)− 2εH 2(εHvi )

= 2H(εHvi )εH − 2εH 2(εHvi )= 0.

Thus, εHgi j ≤ hi j follows from Theorem 4.2. Then hi j ≤ βHgi j follows in thesame way. �

Finally, we state the higher derivative estimate.

Proposition 4.4. There exist constants Cm for m = 1, 2, . . . such that if the secondfundamental form of a complete solution to our evolution equation is bounded by|A| ≤ M up to time t with 0< t ≤ 1/M , then the covariant derivative of the secondfundamental form is bounded by

|∇A| ≤ C1 M/√

t

and the m-th covariant derivative of the second fundamental form is bounded by

|∇m A| ≤ Cm M/tm/2.

Here the norms are taken with respect to the evolving metric.

Proof. By direct calculation, for any m we have an equation

(∂t −4)|∇m A|2 =−2|∇m+1 A|2+

∑i+ j+k=m

∇i A ∗∇ j A ∗∇k A ∗∇m A.

So we can follow the same way using a somewhat standard Bernstein estimate inpartial differential equations to get our theorem; see [Shi 1989] for the argumentin the case of Ricci flow. �

5. Monotonicity formula and long-time behaviors

First, by the positivity of hi j we have H 2/n ≤ |A|2 < H 2. Then from (4-1c) weget

−H 3 < (∂t −4)H ≤−H 3/n.


Thus by maximum principle we obtain

(5-1) 1√2t+1/H 2

min(0)< H(t)≤ 1√

2t/n+1/H 2max(0)

.

With applying maximum principle on (4-1d), we have

|A|2(t)≤ 12t+1/|A|2max(0)

.

Since1

2nt+n/H 2min(0)

< H 2(t)/n ≤ |A|2(t),

we get

(5-2) 12nt+n/H 2

min(0)< |A|2(t)≤ 1

2t+1/|A|2max(0).

In particular, (5-2) implies |A| → 0 as t→+∞. Combining this with our deriva-tives estimate, Proposition 4.4, we conclude the solution of (1-1) exists for all time.

We need the monotonicity formula below to understand the long-time behaviorof the solution to (1-1).

Proposition 5.1. If (gi j (t), hi j (t)) is the solution of (1-1), then we have the formula

∂∂t

∫M

H ndµt =−n(n− 1)∫

M

|∇H |2

H 2 H ndµt − n∫

M

∣∣∣hi j −1n

Hgi j

∣∣∣2 H ndµt .

Proof. It follows from the evolution equations of Proposition 4.1 and direct calcu-lation. �

From Proposition 5.1, we know

(5-3) 0<∫

MH ndµt < C for all t ∈ [0,+∞).

This implies ∫∞

0

∫M

(|∇H |2

H 2 + |hi j −1n

Hgi j |2)

H ndµt dt <∞.

In particular, there is a sequence tk→+∞ such that

tk

∫M

|∇H |2

H 2 H ndµtk → 0 as k→∞,(5-4)

tk

∫M

∣∣∣hi j −1n

Hgi j

∣∣∣2 H ndµtk → 0 as k→∞.(5-5)

162 KUN ZHANG

Let εk = 1/|A|max(tk). We parabolically scale the solution and shift the time tkto the origin 0 by letting

gki j ( · , t)= ε−2

k gi j ( · , tk + ε2k t),

hki j ( · , t)= ε−1

k hi j ( · , tk + ε2k t), where t ∈ [−tk/ε2

k ,+∞).

We can check that (gki j ( · , t), hk

i j ( · , t)) is still a solution to (1-1). From

| Ak( · , t)|2 = |A( · , tk + ε2k t)|2/|A|2max(tk)

and (5-2), it follows that

(5-6) 1/C1 < | Ak( · , t)|2 < C1 for t ∈ [−tk/2ε2k , 0],

where the constant C1 is independent of k.By our derivatives estimate, Proposition 4.4, the uniform bound of the second

fundamental form | Ak( · , t)| implies the uniform bound on all the derivatives ofthe second fundamental form at t = 0 for all k. By Gauss’s equation, we haveuniform bounds of the curvature and all its derivatives at t = 0 for all k.

By (5-3) we know that∫

M(Hk( · , 0))ndµ0<C2. Combining this with (5-1), we

find

(5-7) Vol(M, gki j ( · , 0)) < C3.

On the other hand, by Proposition 4.3, (5-2), and the Gauss equation, we have

(5-8) 0>−1/C4 ≥ Sec(M, gki j ( · , 0))≥−1,

where Sec means sectional curvature.

Theorem 5.2 [Gromov 1978]. Let M be an n-dimensional closed Riemannianmanifold of negative curvature and suppose Sec(M)≥−1. Then

Vol(M)≥{

C(1+ d(M)) if n ≥ 8,C(1+ d1/3(M)) if n = 4, 5, 6, 7,

where we denote by Vol(M) and d(M) the volume and diameter of M , and theconstant C > 0 depends only on n.

Combining (5-8) and (5-7) and this theorem of Gromov, we have

(5-9) diam(M, gki j ( · , 0))≤ C5 and Vol(M, gk

i j ( · , 0))≥ 1/C5.

Now we know (M, gki j ( · , 0), hk

i j ( · , 0)) is a sequence of Riemannian manifoldsthat have uniformly bounded sectional curvature, uniform upper bound on theirdiameters, and uniform lower bound on their volumes. Using Cheeger’s lemmain [Cheeger and Ebin 1975], we have the uniform lower bound of their injectiveradii with respect to gk

i j ( · , 0) for n ≥ 4. Then we can apply the argument used


to prove Hamilton’s compactness theorem of [1995] to extract a convergent sub-sequence (M, gkl

i j ( · , 0), hkli j ( · , 0)) from (M, gk

i j ( · , 0), hki j ( · , 0)). More precisely,

there exists a triple (M∞, g∞i j ( · , 0), h∞i j ( · , 0)) and a sequence of diffeomorphismsfl : M∞ → Ml . Notice that M∞ is diffeomorphic to M , since we have uniformdiameter bound. Also the pull-back metrics ( fl)

∗gkli j ( · , 0) and second fundamental

forms ( fl)∗hkl

i j ( · , 0) converge in the C∞ topology to (g∞i j ( · , 0), h∞i j ( · , 0)).From (5-4) and (5-5) we obtain

tklε−2kl

∫M

|∇ H kl |2(0)

(H kl )2(0)(H kl )n(0)dµtkl

→ 0 as l→∞,

tklε−2kl

∫M

∣∣∣hkli j −

1n

H kl gkli j

∣∣∣2(0)(H kl )n(0)dµtkl→ 0 as l→∞.

Here the norm is taken with respect to gkli j (0). Noting that tklε

−2kl

and |H kl (0)| andVol(M, gkl

i j ( · , 0)) have uniform lower bound, we have

|∇ H kl |(0)→ 0 as l→∞,∣∣∣hkli j −

1n

H kl gkli j

∣∣∣(0)→ 0 as l→∞.

Therefore, by Gauss’s equation, we know Sec(M∞, g∞i j ( · , 0), h∞i j ( · , 0)) is a con-stant equal to −1/n.

Acknowledgments

I am grateful to my advisor Professor B. L. Chen for his guidance.

References

[Cheeger and Ebin 1975] J. Cheeger and D. G. Ebin, Comparison theorems in Riemannian geom-etry, North-Holland Mathematical Library 9, North-Holland, Amsterdam, 1975. MR 56 #16538Zbl 0309.53035

[Ecker 1997] K. Ecker, “Interior estimates and longtime solutions for mean curvature flow of non-compact spacelike hypersurfaces in Minkowski space”, J. Differential Geom. 46:3 (1997), 481–498.MR 98j:58034 Zbl 0909.53045

[Ecker and Huisken 1991] K. Ecker and G. Huisken, “Parabolic methods for the construction ofspacelike slices of prescribed mean curvature in cosmological spacetimes”, Comm. Math. Phys.135:3 (1991), 595–613. MR 92g:53058 Zbl 0721.53055

[Gromov 1978] M. Gromov, “Manifolds of negative curvature”, J. Differential Geom. 13:2 (1978),223–230. MR 80h:53040 Zbl 0433.53028

[Gromov and Thurston 1987] M. Gromov and W. Thurston, “Pinching constants for hyperbolicmanifolds”, Invent. Math. 89:1 (1987), 1–12. MR 88e:53058 Zbl 0646.53037

[Hamilton 1982] R. S. Hamilton, “Three-manifolds with positive Ricci curvature”, J. DifferentialGeom. 17:2 (1982), 255–306. MR 84a:53050 Zbl 0504.53034














http://dx.doi.org/10.1007/BF01404671

http://dx.doi.org/10.1007/BF01404671






164 KUN ZHANG

[Hamilton 1986] R. S. Hamilton, “Four-manifolds with positive curvature operator”, J. DifferentialGeom. 24:2 (1986), 153–179. MR 87m:53055 Zbl 0628.53042

[Hamilton 1995] R. S. Hamilton, “The formation of singularities in the Ricci flow”, pp. 7–136 inSurveys in differential geometry, II (Cambridge, MA, 1993), edited by S.-T. Yau, Int. Press, Cam-bridge, MA, 1995. MR 97e:53075 Zbl 0867.53030

[Huisken 1984] G. Huisken, “Flow by mean curvature of convex surfaces into spheres”, J. Differen-tial Geom. 20:1 (1984), 237–266. MR 86j:53097 Zbl 0556.53001

[Shi 1989] W.-X. Shi, “Deforming the metric on complete Riemannian manifolds”, J. DifferentialGeom. 30:1 (1989), 223–301. MR 90i:58202 Zbl 0676.53044

Received February 25, 2009.

KUN ZHANG

SCHOOL OF MATHEMATICS AND COMPUTATIONAL SCIENCE

SUN YAT-SEN UNIVERSITY

NO. 135 XINGANG ROAD WEST

GUANGZHOU 510275CHINA

[email protected]














GRADIENT ESTIMATES FOR SOLUTIONS OF THE HEATEQUATION UNDER RICCI FLOW

SHIPING LIU

We establish first order gradient estimates for positive solutions of the heatequations on complete noncompact or closed Riemannian manifolds underRicci flows. These estimates improve Guenther’s results by weakening thecurvature constraints. We also obtain a result for arbitrary solutions onclosed manifolds under Ricci flows. As applications, we derive Harnack-type inequalities and second order gradient estimates for positive solutionsof the heat equations under Ricci flow. The results in this paper can beconsidered as generalizing the estimates of Li–Yau and J. Y. Li to the Ricciflow setting.

1. Introduction

In this paper, we mainly generalize Li and Yau’s [1986] and Li’s [1991] gradientestimates to positive solutions of the heat equation under Ricci flow. The Ricciflow,

(1-1) ∂t gi j =−2 Rici j ,

was introduced by Hamilton [1982] to study the Poincare conjecture on compactthree manifolds with positive Ricci curvature. Since then, in the series [1995; 1997;1999], Hamilton created a well-developed theory of Ricci flow as an approachto the Poincare conjecture and the geometrization conjecture. In [2002; 2003],Perelman brought in new ideas and completed the so-called Hamilton program.

Gradient estimates for solutions of the heat equation are very powerful tools inanalysis, as shown for example in [Li 1991; Li and Yau 1986]. Perelman [2002]actually showed a gradient estimate for the fundamental solution of the conjugateheat equation,

1u− Ru+ ∂t u = 0,

under Ricci flow on a closed Riemannian manifold M , where R is the scalar curva-ture. Namely, let u be the fundamental solution of the equation above in M×[0, T ),

MSC2000: primary 58J35, 53C44; secondary 35K55, 53C21.Keywords: gradient estimate, Ricci flow, heat equation, Harnack inequality.

165



166 SHIPING LIU

and let f be the function such that u = (4πτ)−n/2e− f with τ = T − t . Then(τ(21 f − |∇ f |2+ R)+ f − n

)u ≤ 0 in M ×[0, T ).

Equivalently,|∇u|2

u2 −uτu− R ≤ n

τ+

ln uτ+

n2

ln(4πτ)τ

.

This estimate is important in the proof of Perelman. Recently, Kuang and Zhang[2008] established a gradient estimate that works for all positive solutions of theconjugate heat equation under Ricci flow on a closed manifold. (Here and through-out we say that a Riemannian manifold is closed if it is compact without boundary.)As an immediate consequence, they get a Harnack-type inequality. By supposinga lower bound on the Ricci curvature, Zhang [2006] established local gradientestimates for positive solutions of the heat equation under the backward Ricci flow∂t gi j = 2 Rici j on a closed Riemannian manifold. Under stronger curvature con-straints, Guenther [2002] had already established gradient estimates for positivesolutions of the heat equation under Ricci flow on a closed manifold. Using thisresult, she derived a Harnack-type inequality and found a lower bound for the heatkernel under Ricci flow. Here heat kernel means the fundamental solution of theheat equation under Ricci flow, whose existence and basic properties Guenther alsoproved. We weaken her curvature constraints in Section 2 using the method of Liand Yau [1986]. We also get corresponding estimates for complete noncompactmanifolds under Ricci flows. All of these results are generalizations of Li andYau’s gradient estimates.

Interesting in their own right, higher order gradient estimates for heat kernelson complete noncompact Riemannian manifolds under Ricci flows are also closelyrelated with the boundedness of the Riesz transform and the Sobolev inequality.Zhang [2007] found that the noncollapsing result, which is critical in Perelman’sproof of the Poincare conjecture, follows immediately from the Sobolev inequalityunder Ricci flow. Li [1994] used the boundedness of the Riesz transform to provethe Sobolev inequality on Riemannian manifold with some constraints, so it is nat-ural to try to prove with a similar method the Sobolev inequality under Ricci flow.In that method, an important step, completed [Li 1991], is to prove an estimatefor ∇X∇Y H(x, y, t), where H(x, y, t) is the heat kernel and ∇X and ∇Y are thegradient operator in the variables x and y. However, difficulties arise in using thismethod to get the generalization of this estimate under Ricci flow, since in thiscase the heat kernel has properties different from what it had in the fixed metriccase. As a consequence of the first order results in Section 2, in Section 3 we getthe generalization of Li’s [1991] second order gradient estimate for the positivesolution u(x, t) of the heat equation under Ricci flow. Let M be a complete non-compact Riemannian manifold with initial metric g(0). Assume that g(t) evolves

GRADIENT ESTIMATES UNDER RICCI FLOW 167

by Equation (1-1) and its first order covariant derivatives are bounded by k1 and k2.Then we have (the notation is defined in later sections)

|∇2u(x, t)|u(x, t)

+α|∇u(x, t)|2

u2(x, t)− 5α

ut(x, t)u(x, t)

≤ C(k1+ k2/32 + 1/t).

In fact, the estimate for ∇X∇Y H(x, y, t) and the second order gradient estimatefor u(x, t) are proved similarly in [Li 1991]. The main difference is that the latterdoesn’t depend on the special properties of the heat kernel.

For closed Riemannian manifolds under Ricci flows, we get a gradient estimatefor arbitrary solutions of the heat equation at the end of Section 2.

We will use the following notations: We denote by ∇ and 1 the gradient andLaplacian–Beltrami operator under the metric g(t); by C a positive constant thatmay change from line to line; by d(x, y, t) the geodesic distance between x, y ∈Munder g(t); and by ψ(r) a C2 function on [0,+∞), such that

(1-2) ψ(r)={

1 if r ∈ [0, 1],0 if r ∈ [2,+∞),

and

(1-3) 0≤ ψ(r)≤ 1, ψ ′(r)≤ 0, ψ ′′(r)≥−C,|ψ ′(r)|2

ψ(r)≤ C,

where C is an absolute constant. When we say that u(x, t) is a solution to the heatequation, we mean u is a solution that is C2 in x and C1 in t .

2. The first order gradient estimates

In this section, we prove the first order gradient estimates. We will denote

ft(x, t)= ∂t f (x, t)=∂ f (x, t)∂t

for a function f on M × [0, T ], where T is a positive constant. We give a localversion gradient estimate first.

Theorem 1. Let g(t) be a solution to the Ricci flow on a Riemannian manifold Mn

with n≥2 for t in some time interval [0, T ], and suppose−K0≤Ric≤K1 for somepositive constants K0 and K1 and all t ∈[0, T ]. Let M be complete under the initialmetric g(0). Given x0 ∈ M and R > 0, let u be a positive solution to the equation(1− ∂t)u(x, t) = 0 in the cube Q2R,T := {(x, t) | d(x, x0, t) ≤ 2R, 0 ≤ t ≤ T }.Then for (x, t) ∈ Q R,T , we have

(2-1)|∇u(x, t)|2

u2(x, t)−α

ut(x, t)u(x, t)

≤ C(

K1+ K0+1t+

1R2

)for any α > 1, where C depends on n and α only.

168 SHIPING LIU

More explicitly, we have

(2-2)|∇u(x, t)|2

u2(x, t)−α

ut(x, t)u(x, t)

≤nα2

t+

Cα2

R2

(R√

K0+α2

α−1

)+

nα3

α−1K0+ n3/2α2(K0+ K1)+Cα2K1,

for any α > 1, where C depends on n only.

As in the proof in [Li and Yau 1986], let f = log u, and let

F = t(|∇u(x, t)|2

u2(x, t)−α

ut(x, t)u(x, t)

)= t (|∇ f |2−α ft).

Lemma 1. Suppose (M, g(t)) satisfies the hypotheses of Theorem 1. We have

(2-3) (1− ∂t)F ≥−2∇ f · ∇F + tn(|∇ f |2− ft)

2− (|∇ f |2−α ft)

− 2αK0t |∇ f |2− tα2n2(K0+ K1)2.

Proof. For a given time t , choose {x1, x2, . . . , xn} to be a normal coordinate systemat a fixed point. The subscripts i and j will denote covariant derivatives in the xi

and x j directions. We will compute at the fixed point.By a direct computation, we obtain

1F = t(

2∑i, j

f 2i j + 2

∑i, j

fi f j j i + 2∑i, j

Rici j fi f j −α(1 f )t + 2α∑i, j

Rici j fi j

),

where we have used the Ricci identity and the formula

(2-4) 1( ft)= (1 f )t − 2〈Ric,Hess( f )〉.

On the other hand, we have

Ft = (|∇ f |2−α ft)+ t(

2∑i, j

Rici j fi f j + 2∑

i

fti fi −α ft t

).

Then noting that (1− ∂t) f =−|∇ f |2, we arrive at

(2-5) (1− ∂t)F =−2∇ f · ∇F + 2t(∑

f 2i j +α

∑Rici j fi j

)+ 2αt Ric(∇ f,∇ f )− (|∇ f |2−α ft).

Because(Rici j

)n×n is a real symmetric matrix, we obtain

(2-6) −K0− K1 ≤ Rici j ≤ K1+ K0

from −K0≤Ric≤ K1. Applying those bounds and Young’s inequality in the form

|Rici j | | fi j | ≤α2

Ric2i j +

12α

f 2i j ,


we conclude

(2-7) (1− ∂t)F ≥−2∇ f · ∇F + t∑

f 2i j − tα2n2(K0+ K1)

2

− 2αK0t |∇ f |2− (|∇ f |2−α ft).

The lemma is completed with the help of the inequality∑i, j

f 2i j ≥

∑i

f 2i i ≥

1n(1 f )2 = 1

n(|∇ f |2− ft)

2. �

Proof of Theorem 1. Bounded Ricci curvature implies that g(t) is uniformly equiv-alent to the initial metric g(0) (see [Chow et al. 2006, Corollary 6.11]), that is,

e−2K1T g(0)≤ g(t)≤ e2K0T g(0).

By definition, we know that (M, g(t)) is also complete for t ∈ [0, T ].Let

ϕ(x, t)= ϕ(d(x, x0, t))= ψ(

d(x, x0, t)R

)= ψ

(ρ(x, t)

R

),

where ρ(x, t) = d(x, x0, t). For the purpose of applying the maximum principle,the argument of [Calabi 1958] allows us to assume that the function ϕ(x, t), withsupport in Q2R,T , is C2 at the maximum point.

For any 0< T1 ≤ T , let (x1, t1) be the point in Q2R,T1 at which ϕF achieves itsmaximum value. We can assume that this value is positive, because otherwise theproof is trivial. Then at the point (x1, t1), we have

(2-8) ∇(ϕF)= F∇ϕ+ϕ∇F = 0, 1(ϕF)≤ 0, ∂t(ϕF)≥ 0.

Therefore,

(2-9) 0≥ (1− ∂t)(ϕF)

= (1ϕ)F +ϕ(1− ∂t)F −ϕt F + 2∇ϕ∇F.

Using the Laplacian comparison theorem, we have

1ϕ ≥−CR2 −

CR

√K0.

By the evolution formula of the geodesic length under Ricci flow (see [Chow andKnopf 2004]), we calculate

−Fϕt =−Fψ ′( ρ

R

) 1R

dρdt= Fψ ′

( ρR

) 1R

∫γt1

Ric(S, S) ds

≥ Fψ ′( ρ

R

) 1R

K1ρ ≥ Fψ ′( ρ

R

)K1 ≥−F

√C K1,

170 SHIPING LIU

where γt1 is the geodesic connecting x and x0 under the metric g(t1), S is theunit tangent vector to γt1 , and ds is the element of arc length. Substituting the twoinequalities above into (2-9) and using (2-8), we obtain

(2-10) 0≥(−

CR2 −

CR

√K0

)F −√

C K1 F +ϕ(1− ∂t)F.

Applying Lemma 1 to this inequality yields

(2-11) 0≥(−

CR2−

CR

√K0

)F−√

C K1 F−2√

CR√ϕ |∇ f |F+

t1nϕ(|∇ f |2− ft)

2

−ϕ(|∇ f |2−α ft)− 2αK0t1ϕ|∇ f |2− t1α2n2ϕ(K1+ K0)2.

The following computation is almost the same as one in [Li and Yau 1986].Multiplying through by ϕt1 and setting y = ϕ|∇ f |2 and z = ϕ ft , Equation (2-11)becomes

(2-12) 0≥ t1(−

CR2 −

CR

√K0

)(ϕF)−

√C K1t1(ϕF)− 2

√C

Rt21 y1/2(y−αz)

+t21

n(y− z)2− 2αK0t2

1 y−ϕ2 F − t21α

2n2ϕ2(K0+ K1)2.

Using the inequality ax2− bx ≥−b2/(4a) for a, b > 0, one obtains

t21

n(y− z)2− 2

√C

Rt21 y1/2(y−αz)− 2αK0t2

1 y

=t21

n

(1α2 (y−αz)2+

(α−1α

)2y2−2αnK0 y+

(2α−1α2 y− 2n

√C

Ry1/2

)(y−αz)

)≥

t21

n

(1α2 (y−αz)2−

α4n2K 20

(α− 1)2−

α2n2C2(α−1)R2 (y−αz)

).

Hence (2-12) becomes

(2-13) 1nα2 (ϕF)2− (ϕF)

(1+ C

R2 t1+CR

√K0t1+

Cnα2t12(α−1)R2 +

√C K1t1

)−

(nK 2

0α4t2

1

(α− 1)2+ t2

1α2n2ϕ2(K1+ K0)

2)≤ 0.

We apply the quadratic formula and then arrive at

(2-14) ϕF(x1, t1)≤ nα2+

Cnα2

R2

(R√

K0+α2

α−1

)t1+√

Cnα2K1t1

+nα3t1α− 1

K0+ t1ϕn3/2(K0+ K1)α2.

This estimate for ϕF is also correct on {(x, T1) | d(x, x0, T1)≤ 2R} since t1 ≤ T1.Since T1 is arbitrary in 0< T1≤ T , we have completed the proof of Theorem 1. �


The local result above implies a global one.

Corollary 1. Let (M, g(0)) be a complete noncompact Riemannian manifold with-out boundary, and suppose g(t) evolves by Ricci flow in such a way that −K0 ≤

Ric≤K1 for t ∈[0, T ]. Let u be a positive solution to the equation (1−∂t)u(x, t)=0. Then for (x, t) ∈ M × (0, T ], we have

(2-15)|∇u(x, t)|2

u2(x, t)−α

ut(x, t)u(x, t)

≤nα2

t+C(K1+ K0),

for any α > 1, where C depends on n and α only.

Proof. By the uniform equivalence of g(t), we know that (M, g(t)) is completenoncompact for t ∈ [0, T ]. Then let R→+∞ in (2-2). �

Remark 1. When (M, g(0)) is a complete noncompact Riemannian manifold, Shi[1989] gives a sufficient condition for the short time existence of the Ricci flow: Itsuffices that the curvature tensor {Ri jkl} of g(0) satisfies

|Ri jkl |2≤ κ on M,

where 0< κ <+∞ is a constant.

Using Lemma 1, we can also derive a similar gradient estimate on a closedRiemannian manifold.

Theorem 2. Let (M, g(t)) be a closed Riemannian manifold, where g(t) evolvesby Ricci flow in such a way that −K0 ≤ Ric ≤ K1 for t ∈ [0, T ]. If u is a positivesolution to the equation (1− ∂t)u(x, t)= 0, then for (x, t) ∈ M × (0, T ], we have

|∇u(x, t)|2

u2(x, t)−α

ut(x, t)u(x, t)

≤nα2

t+

nα3K0α−1

+ n3/2α2(K0+ K1),(2-16)


Proof. Let notations F and f be as above. Set

F(x, t)= F(x, t)− nα3K0α−1

t − n3/2α2(K0+ K1)t.

If F(x, t)≤ nα2 for any (x, t) ∈ M × (0, T ], then the theorem is proved.If (2-16) doesn’t hold, then at the maximal point (x0, t0) of F(x, t), we have

F(x0, t0) > nα2. Noting F(x, 0) = 0, we know here t0 > 0. Then applying themaximal principle, we have at the point (x0, t0) that

(2-17) ∇F(x0, t0)= 0, 1F(x0, t0)≤ 0, ∂∂t

F(x0, t0)≥ 0.

Therefore we obtain

(2-18) 0≥ (1− ∂t)F ≥ (1− ∂t)F.

172 SHIPING LIU

Using Lemma 1 and the trick in calculating (2-11), we get

(2-19) 0≥t0

nα2

(Ft0

)2−

(Ft0

)−

nα4K 20

(α− 1)2t0− t0α2n2(K0+ K1)

2

+2t0nα−1α2 |∇ f |2 F

t0.

SinceFt0=

Ft0+

nα3K0α−1

+ n3/2α2(K0+ K1) > 0,

we get the inequality

(2-20)t0

nα2

(Ft0

)2−

(Ft0

)−

nα4K 20

(α− 1)2t0− t0α2n2(K0+ K1)

2≤ 0.

Again the quadratic formula gives

(2-21) Ft0≤

nα2

t0+

nα3K0

α− 1+ n3/2α2(K0+ K1).

This implies F(x0, t0)≤ nα2, a contradiction. So (2-16) holds. �

Remark 2. In Corollary 1 and Theorem 2, if K0 = 0, we can let α→ 1.

In fact, the Theorem 2 improves the gradient inequality in [Guenther 2002],which requires the boundedness of the gradient of scalar curvature in addition tothe boundedness of the Ricci curvature. Beginning with this result, we can dothings similar to what was done in [Guenther 2002], such as deriving Harnack-type inequalities.

Corollary 2. Let (M, g(0)) be a complete noncompact Riemannian manifold with-out boundary or a closed Riemannian manifold, and suppose g(t) evolves by Ricciflow for t ∈ [0, T ] in such a way that−K0 ≤Ric≤ K1. If u is a positive solution tothe equation (1− ∂t)u(x, t) = 0, then for any points (x, t1), (y, t2) ∈ M × (0, T ]such that t1 < t2, we have

(2-22) u(x, t1)≤ u(y, t2)( t2

t1

)2nεexp

(∫ 1

0

ε|γ′(s)|2σ2(t2− t1)

ds+Ct2− t1

2ε(K1+ K0)

),

for any ε > 1/2, where C depends on n and ε only, γ(s) is a smooth curve con-necting x and y with γ(1)= x and γ(0)= y, and |γ′(s)|σ is the length of the vectorγ′(s) at time σ = (1− s)t2+ st1.

Proof. First note that the gradient estimates in Corollary 1 and Theorem 2 can bothbe written as

(2-23)|∇u(x, t)|2

u2(x, t)−α

ut(x, t)u(x, t)

≤nα2

t+Cn,α(K1+ K0).


Define l(s)= ln u(γ(s), (1− s)t2+ st1). It is easy to see that l(0)= ln u(y, t2) andl(1)= ln u(x, t1). Direct calculation gives

∂l(s)∂s= (t2− t1)

(∇uu

γ′(s)t2− t1

−ut

u

)≤ε|γ′(s)|2σ2(t2− t1)

+t2− t1

2ε

(C(K1+ K0)+

4ε2nσ(s)

).

Integrating this inequality over γ(s), we have

lnu(x, t1)u(y, t2)

=

∫ 1

0

∂l(s)∂s

ds

≤

∫ 1

0

ε|γ′(s)|2σ2(t2− t1)

ds+Ct2− t1

2ε(K1+ K0)+ 2εn ln(t2/t1). �

We can also get a gradient estimate for an arbitrary solution of the heat equationunder Ricci flow on a closed manifold without any curvature conditions. The aux-iliary function F we take in the following proof is inspired by Hamilton’s [1995]proof of Shi’s [1989] derivative estimates.

Theorem 3. Let (M, g(t)) be a closed Riemannian manifold, where g(t) solvesthe Ricci flow for t ∈ [0, T ]. If u solves 1u− ∂t u = 0, then

(2-24) |∇u(x, t)|2 ≤ 12t

(maxx∈M

u2(x, 0)− u2(x, t))

for (x, t) ∈ M ×[0, T ].

Proof. Since ∂t u =1u, we have

∂t(|∇u|2)= 2 Ric(∇u,∇u)+ 2〈∇u,∇(1u)〉.

Using Bochner’s formula, this becomes

(2-25) ∂t(|∇u|2)=1(|∇u|2)− 2|∇2u|2.

On the other hand,

(2-26) ∂t(u2)=1(u2)− 2|∇u|2.

Let F = t |∇u|2+ Au2, where A is a constant to be fixed. Then combining (2-25)and (2-26) gives

(2-27)

∂t F = |∇u|2+ t (1(|∇u|2)− 2|∇2u|2)+ A(1(u2)− 2|∇u|2)

≤1F + (1− 2A)|∇u|2.

Setting A = 1/2 and applying maximum principle, we conclude

(2-28) F(x, t)≤maxx∈M

F(x, 0)= 12 max

x∈Mu2(x, 0).

This inequality implies the theorem. �

174 SHIPING LIU

Remark 3. For a positive solution of the heat equation u on closed manifoldsunder Ricci flow, Zhang [2006] gives a stronger estimate,

(2-29)|∇u|

u≤

√1t

√ln M

u,

where M = maxx∈M u(x, 0). Similar to the fact that the interpolation inequalityfollows from this estimate in [Zhang 2006], here we get for any x, y ∈ M and0< t ≤ T that

(2-30) u(x, t)≤ u(y, t)+

√C2

d(x, y, t)√

t,

where C =maxx∈M u2(x, 0).

3. The second order gradient estimates

Using Corollary 1, we can generalize to the Ricci flow setting the second ordergradient estimate for the positive solution of the heat equation in [Li 1991].

Theorem 4. Let g(t) be a solution to the Ricci flow on a Riemannian manifold Mn

with n ≥ 2 for t in some time interval [0, T ]. Assume that (M, g(0)) is a completenoncompact manifold without boundary. Suppose the curvature tensor and its firstorder covariant derivatives are bounded throughout by k1 and k2, respectively. Letu be a positive solution to (1− ∂t)u(x, t)= 0. Then for (x, t) ∈ M × (0, T ],

(3-1)|∇

2u(x, t)|u(x, t)

+α|∇u(x, t)|2

u2(x, t)− 5α

ut(x, t)u(x, t)

≤ C(

k1+ k2/32 +

1t

),


To prove the theorem, we set

F(x, y, t)= t F1 = t(|∇

2u(x, t)|u(x, t)

+α|∇u(x, t)|2

u2(x, t)−β

ut(x, t)u(x, t)

),

where β is a constant to be fixed.

Lemma 2. Suppose (M, g(t)) satisfies the hypotheses of Theorem 4. Then forsufficiently small δ > 0 and γ− 1> 0 and some ε > 0, we have

(1− ∂t)F ≥ − 2∇F · ∇ log u+ δαt

F2+ 2δαβF ut

u− 2δα2 F

|∇u|2

u2

−Ck1 F −Ck2

1

4(γ− 1)2t −Ck4/3

2 t − 4βεn2k21 t − F/t

− 2Ct(

4δα3+

δ2α(1−δ)2

)(1t+ k1

)2,


where β = 5α and C depends on n and α.

Proof. As in the proof of Lemma 1, choose {x1, x2, . . . , xn} to be a normal coordi-nate system at a fixed point. Subscripts i , j and k will denote covariant derivativesin the xi , x j and xk directions.

We will calculate the evolution equation for F1. The computation is a little long,so we divide it into three parts.

Part 1. We calculate the equation for |∇2u|/u.It follows from [Li 1991] that

1

(|∇

2u|u

)=

∑u2

i jk +∑

ui j ui jkk

u|∇2u|− 2

∑ui j ui jkuk

u2|∇2u|−

∑k

(∑i j ui j ui jk

)2

u|∇2u|3(3-2)

−|∇

2u|1uu2 + 2

|∇2u||∇u|2

u3 ,

∂t

(|∇

2u|u

)=∂t(|∇

2u|2)2u|∇2u|

−|∇

2u|ut

u2 .(3-3)

Noting the metric evolves by the Ricci flow, we have

∂t(|∇2u|2)= 4

∑Ric(∇u j ,∇u j )+ 2

∑ui j t ui j ,

ui j t = ∂t(ui j −0li j ul)= uti j +

∑l ul(∇i Ric jl +∇ j Ricil −∇l Rici j ).

The Ricci identity gives

ui jkk = ukki j+∑

l

(Rkikl, j ul+ Rkiklul j+ Rk jilulk+ Rk jkluli+ Ri jkl,kul+ Ri jklulk

).

Combining the above and using the Schwarz inequality, we conclude

(1− ∂t)

(|∇

2u|u

)≥−2∇

(|∇

2u|u

)· ∇ log u−Ck1

|∇2u|u−Ck2

|∇u|u.

Part 2. We calculate the equation for |∇u|2/u2. A direct computation shows

(1− ∂t)

(α|∇u|2

u2

)

= 2α

∑u2

i j

u2 + 2α∑

ui u j j i

u2 + 2αRic(∇u,∇u)

u2 − 8α∑

ui j ui u j

u3 − 2α∑

u2i u j j

u3

+ 6α

∑u2

i u2j

u4 − 2αRic(∇u,∇u)

u2 − 2α∑

ui ui t

u2 + 2α∑

u2i ut

u3

= 2α

∑u2

i j

u2 − 8α∑

ui j ui u j

u3 + 6α

∑u2

i u2j

u4

=−2∇(α|∇u|2

u2

)· ∇ log u+ 2α

∑u2

i j

u2 − 4α∑

ui j ui u j

u3 + 2α

∑u2

i u2j

u4 .

176 SHIPING LIU

In the above, the two Ricci curvature terms generated by the Ricci identity and theevolution of the metric are canceled. Applying Young’s inequality, that is,

4α∑

ui j ui u j

u3 ≤ 2(1− δ)α

∑u2

i j

u2 +2α

1−δ

(∑u2

i

u2

)2

for any 0< δ < 1,

we conclude

(1− ∂t)

(α|∇u|2

u2

)≥−2∇

(α|∇u|2

u2

)· ∇ log u+ 2δα

∑u2

i j

u2 −2δα1−δ

(∑u2

i

u2

)2

.

Part 3. Using the formula (2-4) and Young’s inequality, we get for any ε > 0,

(1− ∂t)(β

ut

u

)≥−2∇

(β

ut

u

)· ∇ log u−

β

ε

∑u2

i j

u2 −βε∑

Ric2i j .

Combining Parts 1–3 and using (2-6), we obtain for any 0< δ < 1 and ε > 0

(3-4) (1− ∂t)F1 ≥−2∇F1 ·∇ log u−Ck1|∇

2u|u+

(δα−

β

ε

)∑ u2i j

u2 + δα

∑u2

i j

u2

−2δα1−δ

(∑u2

i

u2

)2

−Ck2|∇u|

u− 4βεn2k2

1 .

By the definition of F1, we have

(3-5)|∇

2u|u≤ F1+β

ut

u,

and∑u2

i j

u2 =

(F1−α

|∇u|2

u2 +βut

u

)2

= F21 +α

2(∑

u2i

u2

)2

+β2 u2t

u2 + 2βF1ut

u− 2αF1

∑u2

i

u2 − 2αβ∑

u2i

u2

ut

u.(3-6)

Inserting (3-5) and (3-6) into (3-4) and applying Young’s inequality to separate themixed items, we arrive at

(3-7) (1− ∂t)F1 ≥−2∇F1 · ∇ log u−Ck1 F1−14Ck2

1/(γ− 1)2

+( 1

2δαβ2−Cβ2(γ− 1)2

)u2t

u2 +

(δα−

β

ε

)∑ u2i j

u2

−

(4δα3+

δ2α(1−δ)2

)(∑ u2i

u2

)2

+ δαF21

+ 2δαβF1ut

u− 2δα2 F1

∑u2

i

u2 −Cδ−1/3k4/32 − 4βεn2k2

1,

for any γ− 1> 0.


Using the inequality

(3-8)(|∇u|2

u2

)2

≤ 2(|∇u|2

u2 − γut

u

)2

+ 2γ2u2t /u

2,

we calculate

(3-9)( 1

2δαβ2−Cβ2(γ− 1)2

)u2t

u2 −

(4δα3+

δ2α(1−δ)2

)(∑u2

i

u2

)2

≥

(12δαβ

2− 2γ2

(4δα3+

δ2α(1−δ)2

)−Cβ2(γ− 1)2

)u2

t

u2

− 2(

4δα3+

δ2α(1−δ)2

)(|∇u|2

u2 − γut

u

)2

.

Setting β = 5α, we check that

12δαβ

2− 2γ2

(4δα3+

δ2α(1−δ)2

)−Cβ2(γ− 1)2

= 8δα3( 2516 − γ

2)−

δ

α(1− δ)2γ2−Cβ2(γ− 1)2

≥ 0 when δ > 0 and γ− 1> 0 are sufficiently small.

Then we can take ε ≥ 5/δ such that δα−β/ε ≥ 0.Consequently, (3-7) becomes

(3-10) (1− ∂t)F1 ≥−2∇F1 · ∇ log u−Ck1 F1−Ck2

1

4(γ− 1)2+ δαF2

1

− 2(

4δα3+

δ2α(1−δ)2

)(|∇u|2

u2 − γut

u

)+ 2δαβF1ut/u− 2δα2 F1

∑u2

i

u2 −Ck4/32 − 4βεn2k2

1 .

We complete the lemma by applying Corollary 1 and noting that

(1− ∂t)F = t (1− ∂t)F1− F1. �

Proof of Theorem 4. As in the proof of Theorem 1, (M, g(t)) is complete fort ∈ [0, T ]. Let

ρ(x, t)= d(x, x0, t) and ϕ(x, t)= ψ(ρ(x, t)/R).

Let ϕF(x, t) := ψ(ρ(x, t)/R)F(x, t), where (x, t) ∈ Q2R,T . Suppose (x1, t1) isthe point where ϕF achieves its maximum in Q2R,T1 , where 0< T1 ≤ T .

178 SHIPING LIU

If |∇2u(x1, t1)| = 0, then by Corollary 1, we have

(ϕF)(x1, t1)= ϕt1

(α|∇u|2

u2 −βut

u

)≤ Cn,α,β (k1t1+ 1) ,(3-11)

which implies (3-1).Using the arguments of [Calabi 1958] and [Li 1991], we can assume that ϕF is

smooth at (x1, t1) and that ϕF(x1, t1) > 0.As in the proof of Theorem 1, at the point (x1, t1), we have

(3-12)0≥ (1− ∂t)(ϕF)

≥

(−

CR2 −

CR

√k1

)F −Ck1 F +ϕ(1X − ∂t)F.

Applying Lemma 2 and Young’s inequality, we obtain for s > 0 that

(3-13) 0≥(−

CR2 −

CR

√k1

)F −Ck1 F −

F |∇ϕ|2

2sϕ− 2Fsϕ

|∇u|2

u2

−Ck1ϕF + δαt1ϕF2− 2Cϕt1

(4δα3+

δ2α(1−δ)2

)(1/t1+ k1)

2

+ 2δαβϕFut/u− 2δα2ϕF∑

u2i

u2 −Ck2

1

4(γ− 1)2ϕt1

−Ck4/32 ϕt1− 4βεn2ϕt1k2

1 −ϕF/t1.

Using Corollary 1, we have

(3-14) 2δαβϕFut/u− 2δα2ϕF∑

u2i

u2 − 2Fsϕ|∇u|2

u2

≥ (2δαβ − 2sγ− 2δα2γ)ϕFut/u−C(2δα2+ 2s)ϕF(1/t1+ K1).

Observe that 2δαβ−2sγ−2δα2γ = 0 when we set s = δα2(4/γ−1). Then (3-13)becomes

(3-15) 0≥ δαt1ϕF2−ϕFC(2s+2δα2) (1/t1+ k1)−

C F2s R2 +

(−

CR2 −

CR

√k1

)F

−Ck1 F −Ck1ϕF −ϕF/t1−Ck2

1

4(γ− 1)2ϕt1− 4βεn2ϕt1k2

1

−Ck4/32 ϕt1− 2Cϕt1

(4δα3+

δ2α(1−δ)2

)(1/t1+ k1)

2.


Multiplying through by ϕt1 and using 0≤ ϕ ≤ 1, we have

(3-16) 0≥ δα(ϕF)2− (ϕF)(( C

R2 +CR

√k1

)t1+ (8Cδα2/γ)(1/t1+ k1)t1

)− (ϕF)

(Ct1

2δα2(4/γ− 1)R2 + (1+Ck1t1))−

Ck21

4(γ− 1)2t21

− 2C(

4δα3+

δ2α(1−δ)2

)(1/t1+ k1)

2t21 − 4βεn2k1t2

1 −Ck4/32 t2

1 .

Applying the quadratic formula, we get

ϕF(x1, t1)≤ C(1+ k1t1+ k2/32 t1+ (C/R2)t1).(3-17)

By an argument similar to one in the proof of Theorem 1, we conclude

F1(x, t)≤ C(k1+ k2/32 + 1/t + 1/R2) in Q R,T ,(3-18)

where C depends on n and α. Because M is noncompact, we can let R→+∞.This completes the proof of Theorem 4. �

Acknowledgment

We thank Professor Jiayu Li for his guidance and encouragement. We also thankYunyan Yang and Jun Sun for their very useful suggestions.

References

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[Chow and Knopf 2004] B. Chow and D. Knopf, The Ricci flow: An introduction, MathematicalSurveys and Monographs 110, American Mathematical Society, Providence, RI, 2004. MR 2005e:53101 Zbl 1086.53085

[Chow et al. 2006] B. Chow, P. Lu, and L. Ni, Hamilton’s Ricci flow, Graduate Studies in Mathemat-ics 77, American Mathematical Society, Providence, RI, 2006. MR 2008a:53068 Zbl 1118.53001

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[Hamilton 1995] R. S. Hamilton, “The formation of singularities in the Ricci flow”, pp. 7–136 inSurveys in differential geometry, II (Cambridge, MA, 1993), edited by S.-T. Yau, International,Cambridge, MA, 1995. MR 97e:53075 Zbl 0867.53030

[Hamilton 1997] R. S. Hamilton, “Four-manifolds with positive isotropic curvature”, Comm. Anal.Geom. 5:1 (1997), 1–92. MR 99e:53049 Zbl 0892.53018

[Hamilton 1999] R. S. Hamilton, “Non-singular solutions of the Ricci flow on three-manifolds”,Comm. Anal. Geom. 7:4 (1999), 695–729. MR 2000g:53034 Zbl 0939.53024



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[Li 1991] J. Y. Li, “Gradient estimate for the heat kernel of a complete Riemannian manifold and itsapplications”, J. Funct. Anal. 97:2 (1991), 293–310. MR 92f:58174 Zbl 0724.58064

[Li 1994] J. Y. Li, “The Sobolev inequality and Sobolev imbedding theorem for Riemannian man-ifolds with nonnegative Ricci curvature”, Chinese Ann. Math. Ser. A 15:4 (1994), 461–471. InChinese. MR 96e:58146

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Received September 10, 2008. Revised September 28, 2008.

SHIPING LIU

ACADEMY OF MATHEMATICS AND SYSTEMS SCIENCE

CHINESE ACADEMY OF SCIENCES

BEIJING 100190CHINA

[email protected]

http://dx.doi.org/10.1016/j.jfa.2008.05.014




http://dx.doi.org/10.1016/0022-1236(91)90003-N

http://dx.doi.org/10.1016/0022-1236(91)90003-N




http://dx.doi.org/10.1007/BF02399203


http://arxiv.org/abs/math.DG/0211159







http://dx.doi.org/10.1093/imrn/rnm056




A 1-COHOMOLOGY CHARACTERIZATION OF PROPERTY (T)IN VON NEUMANN ALGEBRAS

JESSE PETERSON

We obtain a characterization of property (T) for von Neumann algebrasin terms of 1-cohomology, similar to the Delorme–Guichardet theorem forgroups.

0. Introduction

The analogue of group representations in von Neumann algebras is the notion ofcorrespondences which is due to Connes [Connes 1982; 1980; Popa 1986], andhas been a very useful in defining notions such as property (T) and amenability forvon Neumann algebras. It is often useful to view group representations as positivedefinite functions that we obtain through a GNS construction. Correspondencesof a von Neumann algebra N can also be viewed in two separate ways, as HilbertN-N bimodules H, or as completely positive maps φ : N→ N , and the equivalenceof these two descriptions is also realized via a GNS construction. This allows oneto characterize property (T) for von Neumann algebras in terms of completelypositive maps.

For a countable group G there is also a notion of conditionally negative defi-nite functions ψ : G → C, which satisfy ψ(g−1) = ψ(g) and the condition thatfor all n ∈ N, α1, α2, . . . , αn ∈ C and g1, g2, . . . , gn ∈ G, if

∑ni=1 αi = 0 then∑n

i, j=1 α jαiψ(g−1j gi ) ≤ 0. Real-valued conditionally negative definite functions

can be viewed as cocycles b ∈ B1(G, π), where π : G → O(H) is an orthogonalrepresentation of G; see [Bekka et al. 2008]. Real-valued conditionally negativedefinite functions can also be viewed as generators of semigroups of positive def-inite functions by Schoenberg’s theorem. These equivalences then make possiblecertain connections between 1-cohomology, conditionally negative definite func-tions, and positive definite deformations, for example the Delorme–Guichardettheorem [Delorme 1977; Guichardet 1977], which states that a group has prop-erty (T) of Kazhdan [1967] if and only if the first cohomology vanishes for anyunitary representation.

MSC2000: 22D25, 46L10, 46L57.Keywords: finite von Neumann algebras, property (T), closable derivations, completely positive

semigroups.

181



182 JESSE PETERSON

It was Evans [1977] who introduced the notion of bounded conditionally com-pletely positive/negative maps and related them to the study the infinitesimal gen-erators of norm continuous semigroups of completely positive maps. He notedthat this definition gives an analogue to conditionally positive/negative definitefunctions on groups. We will extend the notion of conditionally completely neg-ative maps to unbounded maps and use a GNS type construction to alternatelyview them as closable derivations into a Hilbert N -N bimodule. This is done inthe same spirit as [Sauvageot 1989; 1990], where Sauvageot makes a connectionbetween quantum Dirichlet forms and differential calculus. Indeed, it is shown inTheorem 1.1 that conditionally completely negative maps are in fact extensions ofgenerators associated to completely Dirichlet forms; however we are coming froma different perspective here and so we will present the correspondence betweenconditionally completely negative maps and closable derivations in a way moreclosely related to group theory.

In studying various properties of groups such as property (T) or the Haagerupproperty, one can give a characterization of these properties in terms of bound-edness conditions on conditionally negative definite functions (as, for example in[Akemann and Walter 1981]); hence one would hope that this is possible for vonNeumann algebras as well.

We will show that one can indeed obtain a characterization of property (T) in thisway. The main result is that a separable finite factor has property (T) if and onlyif the 1-cohomology spaces of closable derivations vanish whenever the domaincontains a non-0 set (see Section 3 for the definition of a non-0 set).

Theorem 0.1. Suppose that N is a separable finite factor. Then the followingconditions are equivalent:

(1) N has property (T).

(2) N does not have property 0, and given any weakly dense ∗-subalgebra N0 ⊂

N with 1 ∈ N0 such that N0 contains a non-0 set, every densely defined clos-able derivation on N0 into a Hilbert N-N bimodule is inner.

(3) There exists a weakly dense ∗-subalgebra N0 ⊂ N such that N0 is countablygenerated as a vector space and every closable derivation into a Hilbert N-Nbimodule whose domain contains N0 is inner.

This is the analogue to the Delorme–Guichardet theorem for groups. As acorollary we obtain that if X1, . . . , Xn generate a finite factor with property (T),and if at least one of the X j has diffuse spectrum, then the derivations ∂X i from[Voiculescu 1998] cannot all be closable, and hence the conjugate variables cannotall exist in L2(N , τ ).

1-COHOMOLOGY OF PROPERTY (T) IN VON NEUMANN ALGEBRAS 183

Corollary 0.2. Suppose that N is a finite factor with property (T). Let X1, . . . , Xn

generate N as a von Neumann algebra such that C[X1, . . . , Xn] contains a non-0set in the sense of Definition 3.1. Then 8∗(X1, . . . , Xn)=∞.

We also give an application showing that man amalgamated, free products offinite von Neumann algebras do not have property (T).

Theorem 0.3. Let N1 and N2 be finite von Neumann algebras with normal faithfultracial states τ1 and τ2 respectively, and suppose that B is a common von Neumannsubalgebra such that τ1|B = τ2|B . If there are unitaries ui ∈ U(Ni ) such thatEB(ui )= 0 for i = 1, 2, then M = N1 ∗B N2 does not have property (T).

Other than the introduction there are four sections. Section 1 establishes thedefinitions and notations and gives the connection between closable derivations,conditionally completely negative maps, and semigroups of completely positivemaps. In Section 2, we characterize when a closable derivation is inner, in termsof the conditionally completely negative map and the semigroup. In Section 3,we state and prove the main theorem, Theorem 3.2, and in Section 4, we give theapplication to amalgamated free products (Corollary 4.2).

1. A GNS-type construction

1.1. Conditionally completely negative maps. Let N be a finite von Neumannalgebra with normal faithful trace τ .

Definition. Suppose 9 : N → L1(N , τ ) is a ∗-preserving linear map whose do-main is a weakly dense ∗-subalgebra D9 of N such that 1 ∈ D9 . Then 9 is aconditionally completely negative (c.c.n.) map on N if,

(1.1.1) for all n ∈ N, x j , y j ∈ D9, and j ≤ n,n∑

j=1

x j y j = 0 impliesn∑

i, j=1

y∗j9(x∗

j xi )yi ≤ 0.

It is not hard to see that condition (1.1.1) can be replaced with the condition that

(1.1.1′) for all n ∈ N, x j , y j ∈ D9, and j ≤ n,n∑

j=1

x j y j = 0 impliesn∑

i, j=1

τ(9(x∗j xi )yi y∗j )≤ 0.

If φ : N → N is completely positive and k ∈ N , then 9(x)= k∗x + xk−φ(x)gives a map that is c.c.n. and bounded. If δ : N→ L2(N , τ ) is a derivation, then δis c.c.n. Also if 9 is a c.c.n. map and α : N→ N is a τ -preserving automorphism,then 9 ′ = α ◦9 ◦α−1 is another c.c.n. map.

184 JESSE PETERSON

One can check that if 91 and 92 are c.c.n. such that D91∩D92 is weakly densein N , and if s, t ≥ 0, then 9 = s91 + t92 is c.c.n. Also if {9t }t is a family ofc.c.n. maps on the same domain and 9 is the pointwise ‖ · ‖1-limit of {9t }t , then9 is c.c.n.

We say that 9 is symmetric if τ(9(x)y)= τ(x9(y)) for all x, y ∈ D9 . We saythat 9 is conservative if τ ◦9 = 0. We also say that 9 is closable if the quadraticform q on L2(N , τ ) given by D(q) = D9 and q(x) = τ(9(x)x∗) is closable.Note that we will see in Section 1.3 that if 9 : D9 → L2(N , τ ) ⊂ L1(N , τ ) is aconservative symmetric c.c.n. map, then 9 is automatically closable.

If 9 is a conservative symmetric c.c.n. map, then τ(9(1)x)= τ(9(x))= 0 forall x ∈ D9 ; hence 9(1)= 0. Also, if 9 is symmetric and 9(1)≥ 0, then given anyx ∈ D9 , if we let x1 = x , x2 = 1, y1 =−1, and y2 = x , then the above conditionimplies that τ(9(x)x∗) ≥ 0, so that we actually have positivity instead of just thesymmetry condition.

1.2. Closable derivations. Let H be a Hilbert N -N bimodule. A derivation of Nis a (possibly unbounded) map δ : N→H defined on a weakly dense ∗-subalgebraDδ of N such that 1 ∈ Dδ, and such that δ(xy)= xδ(y)+ δ(x)y for all x, y ∈ Dδ.The map δ is closable if it is closable as an operator from L2(N , τ ) to H.

The map δ is inner if δ(x) = xξ − ξ x for some ξ ∈ H. It is spanning ifspDδδ(Dδ)=H, and it is real if

〈xδ(y), δ(z)〉H = 〈δ(z∗), δ(y∗)x∗〉H for all x, y, z ∈ Dδ.

If δ′ : Dδ→H′ is another derivation, then we say that δ and δ′ are equivalent ifthere exists a unitary map U :H→H′ such that U (xδ(y)z)= xU (δ(y))z= xδ′(y)zfor all x, y, z ∈ Dδ.

Recall that if H is a Hilbert N -N bimodule, then we can define the adjointbimodule H◦, where H◦ is the conjugate Hilbert space of H and the bimodulestructure is given by xξ ◦y = (y∗ξ x∗)◦. If δ : Dδ → H is a closable derivation,then we may define the adjoint derivation δ◦ : Dδ→H◦ by setting δ◦(x)= δ(x∗)◦;then δ◦ is a closable derivation and the derivations 1

2(δ+δ◦) and 1

2(δ−δ◦) are real

derivations from Dδ to H⊕H◦.

1.3. From conditionally completely negative maps to closable derivations. Let9 be a conservative symmetric c.c.n. map on N with domain D9 . We associate to9 a derivation in the following way (compare with [Sauvageot 1989]):

Let

H0 =

{ n∑i=1

xi ⊗ yi ∈ D9 ⊗ D9

∣∣∣ n∑i=1

xi yi = 0}.


Define a sesquilinear form on H0 by

⟨ n∑i=1

x ′i ⊗ y′i ,m∑

j=1

x j ⊗ y j

⟩9=−

12

n∑i=1

m∑j=1

τ(9(x∗j x ′i )y′

i y∗j ).

The positivity of 〈 · , · 〉9 is equivalent to the c.c.n. condition on 9. Let H be theclosure of H0 after we mod out by the kernel of 〈 · , · 〉9 . If p=

∑nk=1 xk⊗ yk such

that∑n

k=1 xk yk = 0, then

x 7→ −12

n∑i, j=1

τ(x∗j xxi9(yi y∗j )) and y 7→ −12

n∑i, j=1

τ(9(x∗j xi )yi yy∗j )

are both positive normal functionals on N with norm 〈p, p〉9 . We also have leftand right commuting actions of D9 on H0 given by

xpy = x( n∑

k=1

xk ⊗ yk

)y =

n∑k=1

(xxk)⊗ (yk y),

and by the preceding remarks we have, for all x, y ∈ D9 ,

〈xp, xp〉9 = 〈x∗xp, p〉9 ≤ ‖x∗x‖〈p, p〉9 = ‖x‖2〈p, p〉9,

〈py, py〉9 ≤ ‖y‖2〈p, p〉9 .

Hence the above actions of D9 pass to commuting left and right actions on H, andthey extend to left and right actions of N on H given by the formulas

⟨x[∑n

i=1 x ′i ⊗ y′i],[∑m

j=1 x j ⊗ y j]⟩

H=

n∑i=1

m∑j=1

τ(x∗j xx ′i9(y′

i y∗j )),

⟨[∑ni=1 x ′i ⊗ y′i

]y,[∑m

j=1 x j ⊗ y j]⟩

H=

n∑i=1

m∑j=1

τ(9(x∗j x ′i )y′

i yy∗j ).

Since the above forms are normal, the left and right actions commute and arenormal, thus making H into a Hilbert N -N bimodule.

Define δ9 : D9→H by x 7→ [x⊗1−1⊗ x]. Then δ9 is a derivation such that,for all x, y ∈ D9 ,

〈δ9(x), δ9(y)〉H = 〈x ⊗ 1− 1⊗ x, y⊗ 1− 1⊗ y〉9

=−12τ(9(y

∗x))+ 12τ(9(x)y

∗)+ 12τ(9(y

∗)x)− 12τ(9(1)xy∗)

= τ(9(x)y∗).

186 JESSE PETERSON

Also δ9 is real since, for all x, y, z ∈ D9 ,

〈xδ9(y), δ9(z)〉H = 〈xy⊗ 1− x ⊗ y, z⊗ 1− 1⊗ z〉9

=−12τ(9(z

∗xy))+ 12τ(9(xy)z∗)+ 1

2τ(9(z∗x)y)− 1

2τ(9(x)yz∗)

=−12τ(9(1)z

∗xy)+ 12τ(9(z

∗)xy)+ 12τ(9(y)z

∗x)− 12τ(9(yz∗)x)

= 〈1⊗ z∗− z∗⊗ 1, 1⊗ y∗x∗− y∗⊗ x∗〉9 = 〈δ9(z∗), δ9(y∗)x∗〉H.

It follows that δ9 is closable if 9 is. Also if 9 : D9→ L2(N , τ )⊂ L1(N , τ ), thenwe would have D9 = D(δ∗9δ9), which would show that δ (and hence also 9) isclosable.

We will also assume that δ9 is spanning by restricting to spD9δ(D9)⊂H.It is not really much of a restriction that 9(1)= 0, since if 9 is any symmetric

c.c.n. map with 9(1)∈ L2(N , τ ), then 9 ′(x)=9(x)− 129(1)x−

12 x9(1) defines

a symmetric c.c.n. map with 9 ′(1)= 0.

1.4. From closable derivations to conditionally completely negative maps. LetH be a Hilbert N -N bimodule, and suppose that δ : N → H is a closable realderivation defined on a weakly dense ∗-subalgebra Dδ of N with 1 ∈ Dδ.

Define

D9 ={

x ∈ D(δ)∩ N | y 7→ 〈δ(x), δ(y∗)〉 gives a normal linear functional on N}.

Then by [Sauvageot 1990; Davies and Lindsay 1992], D(δ)∩N is a ∗-subalgebra,and hence one can show that D9 is a dense ∗-subalgebra of N . We define the map9δ : D9 → L1(N , τ ) by letting 9δ(x) be the Radon–Nikodym derivative of thenormal linear functional y 7→ 〈δ(x), δ(y∗)〉. Since δ is closable, 9δ is also.

Since δ is real, 9δ is a symmetric ∗-preserving map such that τ ◦9 = 0, and ifn ∈ N and x1, x2, . . . , xn, y1, yx , . . . , yn ∈ D9 such that

∑ni=1 xi yi = 0, then

n∑i, j=1

τ(9(x∗j xi )yi y∗j )=n∑

i, j=1

〈δ(x∗j xi ), δ(y j y∗i )〉H

=

n∑i, j=1

〈x∗j δ(xi ), y jδ(y∗i )+ δ(y j )y∗i 〉H+〈δ(x∗

j )xi , y jδ(y∗i )+ δ(y j )y∗i 〉H

=

n∑i, j=1

〈δ(xi )yi , x jδ(y j )〉H+〈xiδ(yi ), δ(x j )y j 〉H =−2∥∥∥ n∑

i=1

δ(xi )yi

∥∥∥2

H≤ 0.

Hence 9δ is a conservative symmetric c.c.n. map on D9 .Note that if we restrict ourselves to closable derivations that are spanning, then

an easy calculation shows that the constructions above are inverses of each otherin the sense that 9δ9 |D9 =9 and δ9δ ∼= δ.


1.5. Closable derivations and c.c.n. maps from groups. Let 0 be a discretegroup, (C, τ0) a finite von Neumann algebra with a normal faithful trace, andσ a cocycle action of 0 on (C, τ0) by τ0-preserving automorphisms. Denoteby N = C × σ0 the corresponding cross-product algebra with trace τ given byτ(6cgug) = τ0(ce), where cg ∈ C and {ug}g ⊂ N denote the canonical unitariesimplementing the action σ on C .

Let (π0,H0) be a unitary or orthogonal representation of 0, and let b : 0→H0

be an (additive) cocycle of 0, that is, b(gh) = π0(g)b(h)+ b(g) for all g, h ∈ 0.Set Hπ0 to be the Hilbert space H0⊗RL2(N , τ ) if π0 is an orthogonal representationand H0⊗CL2(N , τ ) if π0 is a unitary representation. We let N act on the rightof Hπ0 by (ξ ⊗ x)y = ξ ⊗ (x y) for x, y ∈ N and ξ ∈ H0 and on the left byc(ξ ⊗ x)= ξ ⊗ (cx) and ug(ξ ⊗ x)= (π0(g)ξ)⊗ (ugx) for c ∈ C , x ∈ N , g ∈ 0,and ξ ∈H0. Let D0 be the ∗-subalgebra generated by C and {ug}g. We define δb

by δb(cgug)= cgδb(ug)= b(g)⊗ cgug for cg ∈C and g ∈0; then we can extend δb

linearly so that δb is a derivation on D0. If (π0,H0) is an orthogonal representationand 1g denotes the Dirac delta function at g, then

〈cugδb(uh), δb(uk)〉 = 〈π0(g)b(h), b(k)〉〈cuguh, uk〉

= 〈−π0(g)π0(h)b(h−1),−π0(k)b(k−1)〉〈cuguh, uk〉1k(gh)

= 〈b(k−1), b(h−1)〉〈u∗k , u∗hu∗gc∗〉

= 〈δb(u∗k), δb(u∗h)u∗

gc∗〉,

for all g, h, k ∈ 0 and c ∈ C , thus showing that δb is real.Also we have∣∣〈δb(cgug), δb(

∑h∈0 dhuh)〉

∣∣= ∣∣∑h∈0〈b(g), b(h)〉〈cgug, dhuh〉∣∣

= ‖b(g)‖2∣∣〈cgug,

∑h∈0 dhuh〉

∣∣≤ ‖b(g)‖2‖cg‖

∥∥∑h∈0 dhuh

∥∥1,

for all g ∈ 0, cg ∈ C and∑

h∈0 dhuh ∈ D0. Hence if x =∑

g∈0 cgug ∈ D0 andy ∈ D0, then |〈δb(x), δb(y)〉| ≤ (

∑g∈0‖b(g)‖

2‖cg‖)‖y‖1. In particular this shows

that δb is closable.Now suppose that ψ : 0 → C is a real-valued conditionally negative definite

function on 0 such that ψ(e) = 0, and let (πψ , bπ ) be the representation andcocycle that correspond to ψ through the GNS construction [Bekka et al. 2008].Let (H, δ) denote the Hilbert N -N bimodule and closable derivation constructedout of (πψ , bπ ) as above, and let9 be the symmetric c.c.n. map associated to (H, δ)as in Section 1.4. Then a calculation shows that9(

∑g cgug)=

∑g ψ(g)cgug, and

in fact it is easy to show that this equation still describes a c.c.n. map even if ψ isnot real valued.

188 JESSE PETERSON

Conversely, if (H, δ) is a Hilbert N -N bimodule and a closable derivation suchthat δ is defined on the ∗-subalgebra generated by C and {ug}g, then we can asso-ciate to it a representation π0 on H0 = sp{δ(ug)u∗g | g ∈ 0} by π0(g)ξ ′ = ugξ

′u∗gfor ξ ′ ∈H0. Also we may associate to δ a group cocycle b on 0 by b(g)= δ(ug)u∗gfor g ∈ 0. If 9 is a c.c.n. map that is also defined on the ∗-subalgebra generatedby C and {ug}g, then we can associate to it a conditionally negative definite func-tion ψ by ψ(g) = τ(9(ug)u∗g). Furthermore if δ is real, then by taking only thereal span above, we see that H0 is a real Hilbert space and π0 is an orthogonalrepresentation; also ψ is real valued if and only if 9 is symmetric, and if (H, δ)and 9 correspond to each other as in Sections 1.3 and 1.4, then (π0, b) and ψcorrespond to each other via the GNS construction.

1.6. Examples from free probability. We now have two main examples of clos-able derivations, those that are inner, and those that come from cocycles on groups.Voiculescu [1998; 1999] uses certain derivations in a key role for his nonmicro-states approach to free entropy and mutual free information. These derivations willgive us more examples of closable derivations under certain circumstances.

1.6.1. The derivation ∂X from [Voiculescu 1998]. Let B ⊂ N be a ∗-subalgebrawith 1 ∈ B and X = X∗ ∈ N . If we denote by B[X ] the subalgebra generated byB and X , and if X and B are algebraically free (that is, they do not satisfy anynontrivial algebraic relations), then there is a well-defined unique derivation

∂X : B[X ] → B[X ]⊗ B[X ] ⊂ L2(N , τ )⊗ L2(N , τ )

such that ∂X (X)= 1⊗ 1 and ∂X (b)= 0 for all b ∈ B.We note that if ∂X is inner, then identifying L2(N , τ ) ⊗ L2(N , τ ) with the

Hilbert–Schmidt operators gives the existence of a Hilbert–Schmidt operator thatcommutes with B. Therefore if B contains a diffuse element (that is, one generatinga von Neumann algebra without minimal projections), then ∂X is not inner.

From [Voiculescu 1998], the conjugate variable J (X :B) of X with respect to Bis an element in L1(W ∗(B[X ]), τ ) such that τ(J (X :B)m)= τ ⊗ τ(∂X (m)) for allm ∈ B[X ], that is, J (X :B)= ∂∗X (1⊗ 1).

If J (X :B) exists and is in L2(N , τ ) (as in the case when we perturb a setof generators by free semicircular elements), then ∂X is a closable derivation by[Voiculescu 1998, Corollary 4.2]

1.6.2. The derivation δA:B from [Voiculescu 1999]. Suppose A, B ⊂ N are two∗-subalgebras with 1 ∈ A, B. If we denote by A ∨ B the subalgebra generatedby A and B, and if A and B are algebraically free, then we may define a uniquederivation

δA:B : A∨ B→ (A∨ B)⊗ (A∨ B)⊂ L2(N , τ )⊗ L2(N , τ )


by δA:B(a)= a⊗ 1− 1⊗ a for all a ∈ A, and δA:B(b)= 0 for all b ∈ B.For the same reason as above, if B contains a diffuse element and A 6= C, then

the derivation is not inner.Recall from [Voiculescu 1999] that the liberation gradient j (A :B) of (A, B) is

an element in L1(W ∗(A∪ B), τ ) such that τ( j (A :B)m)= τ ⊗ τ(δA:B(m)) for allm ∈ A∨ B, that is, j (A :B)= δ∗A:B(1⊗ 1).

If j (A :B) exists and is in L2(N , τ ), then δA:B is a closable derivation by[Voiculescu 1999, Corollary 6.3]

1.7. Generators of completely positive semigroups. Suppose N is a finite vonNeumann algebra with normal faithful trace τ . A weak*-continuous semigroup{φt }t≥0 on N is said to be symmetric if τ(xφt(y)) = τ(φt(x)y) for all x, y ∈ N ,and completely Markovian if each φt is a unital c.p. map on N . We denote by 1the generator of a symmetric completely Markovian semigroup {φt }t≥0 on N , thatis, 1 is the densely defined operator on N described by

D(1)= {x ∈ N : (x −φt(x))/t has a weak limit as t→ 0},

and 1(x)= limt→0(x−φt(x))/t . We also let 1 denote the generator of the corre-sponding semigroup on L2(N , τ ). Then 1 describes a completely Dirichlet form[Davies and Lindsay 1992] on L2(N , τ ) by

D(E)= D(11/2) and E(x)= ‖11/2(x)‖22.

From [Davies and Lindsay 1992], D(E)∩N is a weakly dense ∗-subalgebra, andhence it follows from [Sauvageot 1989] that there exists a Hilbert N -N bimodule H

and a closable derivation δ : D(E) ∩ N → H such that E(x) = ‖δ(x)‖2 for allx ∈ D(E) ∩ N . Conversely it follows from [Sauvageot 1990] that if D(δ) is aweakly dense ∗-subalgebra with 1∈D(δ) and δ :D(δ)→H is a closable derivation,then the closure of the quadratic form given by ‖δ(x)‖2 is completely Dirichlet onL2(N , τ ) and hence generates a symmetric completely Markovian semigroup asabove (see also [Cipriani and Sauvageot 2003]).

From Sections 1.3 and 1.4 and from the remarks above, we obtain the following.

Theorem 1.1. Let N0 ⊂ N be a weakly dense ∗-subalgebra with 1 ∈ N0, andsuppose9 : N0→ L1(N , τ ) is a closable, conservative, symmetric c.c.n. map suchthat 9−1(L2(N , τ )) is weakly dense in N. Then 1=9|9−1(L2(N ,τ )) is closableas a densely defined operator on L2(N , τ ), and 1 is the generator of a symmetriccompletely Markovian semigroup on N. Conversely, if1 is the generator of a sym-metric completely Markovian semigroup on N , then 1 extends to a conservative,symmetric c.c.n. map9 : N0→ L1(N , τ ), where N0 is the ∗-subalgebra generatedby D(1).

190 JESSE PETERSON

2. A characterization of inner derivations

Let N be a finite von Neumann algebra with normal faithful trace τ . Given asymmetric c.c.n. map 9 on N , we will now give a characterization of when 9 isnorm bounded.

Theorem 2.1. Let 9 : D9 → L1(N , τ ) be a closable, conservative, symmetricc.c.n. map with weakly dense domain D9 . Let δ : D9→H be the closable deriva-tion associated with 9. Then the following conditions are equivalent:

(a) δ extends to an everywhere-defined derivation δ′ that is inner and such that forany x ∈ N there exists a constant Cx > 0 such that |〈δ′(x), δ′(y)〉| ≤ Cx‖y‖1for all y ∈ N.

(b) There exists a constant C > 0 such that |〈δ(x), δ(y)〉| ≤ C‖x‖‖y‖1 for allx, y ∈ D9 .

(c) 9 is norm bounded on (D9)1.

(d) The image of 9 is contained in N ⊂ L1(N , τ ), and −9 extends to a mappingthat generates a norm continuous semigroup of normal c.p. maps.

(e) There exists a k ∈ N and a normal c.p. map φ : N→ N with the property that9(x)= k∗x + xk−φ(x) for all x ∈ D9 .

Proof. (a) implies (c): Let δ′ be the everywhere-defined extension of δ, and let9 ′ bethe c.c.n. map associated with δ′. Since for any x ∈ N there exists a constant Cx >0such that |〈δ′(x), δ′(y)〉|≤Cx‖y‖1 for all y ∈ N , the image of9 ′ is contained in N .Also since 9 ′(1)= 0 we have 9 ′(x∗x)− x∗9 ′(x)−9 ′(x∗)x ≤ 0 for all x ∈ D9 ′ ,and so −9 ′ is a dissipation [Lindblad 1976; Kishimoto 1976]. Since −9 ′ is alsoeverywhere-defined, it is bounded by [Kishimoto 1976, Theorem 1].

(b) is equivalent to (c): If (b) holds, then for all x, y ∈ D9 ,

|τ(9(x)y∗)| = |〈δ(x), δ(y)〉| ≤ C‖x‖‖y‖1.

So by taking the supremum over all y ∈ D9 such that ‖y‖1 ≤ 1, we find that‖9(x)‖ ≤ C‖x‖ for all x ∈ D9 .

Suppose now 9 is bounded by C > 0. Then for all x, y ∈ D9 ,

|〈δ(x), δ(y)〉| = |τ(9(x)y∗)| ≤ ‖9(x)‖‖y‖1 ≤ C‖x‖‖y‖1.

(c) implies (d): This follows from [Evans 1977, Proposition 2.10].

(d) implies (e): This is [Christensen and Evans 1979, Theorem 3.1].

(e) implies (a): Suppose that for k ∈ N and φ c.p., we have 9(x) = k∗x + xk −φ(x) for all x ∈ N . Let φ′ = τ(φ(1))−1φ, and let (H, ξ) be the pointed HilbertN -bimodule associated with φ′. Hence if we set δ′(x)= (τ (φ(1))/2)1/2[x, ξ ], then


we have δ′ ∼= δ. By replacing k with 12(k + k∗) and φ with 1

2(φ + φ∗), we may

assume that φ is symmetric; it is then easy to verify that there exists a constantC > 0 such that |〈δ′(x), δ′(y)〉| ≤ C‖x‖‖y‖1 for all x, y ∈ N . Hence δ′ gives aneverywhere-defined extension of δ satisfying the required properties. �

Our next result is in the same spirit as Theorem 2.1. It provides several equiva-lent conditions for a closable derivation to be inner.

Theorem 2.2. Let 9 : D9 → L1(N , τ ) be a closable, conservative, symmetricc.c.n. map with weakly dense domain D9 . Let δ : D9→H be the closable deriva-tion associated with 9. Then the following conditions are equivalent:

(α) δ is inner.

(β) δ is bounded on (D9)1.

(γ) 9 is ‖ · ‖1-bounded on (D9)1.

(δ) 9 can be approximated uniformly by c.p. maps in the sense that, for all ε > 0,there exists a k ∈ N and a normal c.p. map φ such that

‖9(x)− k∗x − xk+φ(x)‖1 ≤ ε‖x‖ for all x ∈ D9 .

Proof. (α) implies (δ): Suppose there is a ξ ∈ H such that δ(x) = xξ − ξ x forall x ∈ D9 . Let ε > 0. Since the subspace of “left and right bounded” vectors isdense in H, we may choose ξ0 ∈H so that there exists a constant C > 0 such that‖xξ0‖ ≤ C‖x‖2 for all x ∈ N , ‖ξ0‖ ≤ ‖ξ‖, and also ‖ξ − ξ0‖< ε/8‖ξ‖. Since ξ0

is “bounded”, we may let φξ0 be the normal c.p. map associated with ξ0/‖ξ0‖. Letφ = 2‖ξ0‖

2φξ0 , and let k = φ(1)/2.Note that since δ is real, ξ0 is real also, that is, 〈xξ0, ξ0 y〉 = 〈y∗ξ0, ξ0x∗〉 for

all x, y ∈ N .Then if x, y ∈ D9 , we have

τ((9(x)− k∗x − xk+φ(x))y∗)

= τ(9(x)y∗)− 12τ(φ(1)xy∗)− 1

2τ(xφ(1)y∗)+ τ(φ(x)y∗)

= 〈δ(x), δ(y)〉− 〈xy∗ξ0, ξ0〉− 〈y∗xξ0, ξ0〉+ 2〈xξ0 y∗, ξ0〉

= 〈xξ − ξ x, yξ − ξ y〉− 〈xξ0− ξ0x, yξ0− ξ0 y〉.

Hence,

|〈9(x)− k∗x − xk+φ(x), y〉|

≤ ‖xξ − ξ x‖‖yξ − ξ y− yξ0+ ξ0 y‖+‖yξ0− ξ0 y‖‖xξ − ξ x − xξ0+ ξ0x‖

≤ 4‖x‖‖ξ‖‖y‖‖ξ − ξ0‖+ 4‖y‖‖ξ0‖‖x‖‖ξ − ξ0‖

≤ ε‖x‖‖y‖.

The desired result follows by taking the supremum over all y ∈ (D9)1.

192 JESSE PETERSON

(δ) implies (γ): Let k ∈ N and φ c.p. such that ‖9(x)−k∗x− xk+φ(x)‖1 ≤ ‖x‖.By [Popa 2006, 1.1.2], ‖φ(x)‖2≤‖φ(1)‖2‖x‖ for all x ∈ N . Hence, for all x ∈D9 ,

‖9(x)‖1 ≤ ‖9(x)− k∗x − xk+φ(x)‖1+‖k∗x − xk+φ(x)‖2

≤ (1+ 2‖k‖2+‖φ(1)‖2)‖x‖.

Thus 9 is bounded in ‖·‖1 on (D9)1.

(γ) implies (β): If ‖9(x)‖1 ≤ C‖x‖ for all x ∈ D9 , then

‖δ(x)‖2 = τ(9(x)x∗)≤ ‖9(x)‖1‖x‖ ≤ C‖x‖2 for all x ∈ D9 .

(β) implies (α): Since δ is bounded on (D9)1, we may extend δ to a derivation onthe C∗-algebra A that is generated by D9 . Let X = {δ(u)u∗ | u ∈U(A)}. For eachv ∈U(A), we let v act on H by v ·ξ = vξ+δ(v). Let ξ0 be the center of the set X .Then since ‖v · ξ −v ·η‖ = ‖ξ −η‖ = ‖ξv−ηv‖ for all ξ, η ∈H, the center of theset v · X is v · ξ0, and the center of the set Xv is ξ0v. Further we have v · X = Xv,and thus v ·ξ0= ξ0v. Since v was arbitrary and every x ∈ A is a linear combinationof unitaries, we have δ(x)= ξ0x − xξ0 for all x ∈ A. �

In general, 9 may be unbounded in ‖ · ‖1 even if ‖φt(x)− x‖2 converges to 0uniformly on N1. However, the next section shows that this cannot happen if Nhas property (T) and the domain of 9 contains a “critical set” as in [Connes andJones 1985, Proposition 1].

3. Property (T) in terms of closable derivations

Suppose M is a finite von Neumann algebra with countable decomposable center.We will say that M has property (T) if the inclusion (M ⊂ M) is rigid in the senseof [Popa 2006]; that is, M has property (T) if and only if there exists a normalfaithful tracial state τ ′ on M such that one of these equivalent conditions hold:

• For all ε > 0, there exists a finite F ′ ⊂ M and a δ′ > 0 such that if H is aHilbert M-M bimodule with a vector ξ ∈H satisfying the conditions

‖〈 · ξ, ξ〉− τ ′‖ ≤ δ′, ‖〈ξ · , ξ〉− τ ′‖ ≤ δ′, ‖yξ − ξ y‖ ≤ δ′ for all y ∈ F ′,

then there exists a ξ0 ∈H such that ‖ξ0− ξ‖ ≤ ε and xξ0 = ξ0x for all x ∈ M .

• For all ε > 0, there exists a finite F ⊂ M and a δ > 0 such that if φ : M→ Mis a normal, completely positive map with

τ ′ ◦φ ≤ τ ′, φ(1)≤ 1, ‖φ(y)− y‖2 ≤ δ for all y ∈ F,

then ‖φ(x)− x‖2 ≤ ε for all x ∈ M with ‖x‖ ≤ 1.


Furthermore, Popa [2006] showed that the above definition is independent of thetrace τ ′, and in the case when N is a factor, this agrees with the original definitionin [Connes and Jones 1985].

In this section we will obtain a characterization of property (T) in terms of certainboundedness conditions on c.c.n. maps. Since we are dealing with unboundedmaps, the domain of a map will be of crucial importance. We will thus want toconsider c.c.n. maps whose domain contains a “critical set”, which by [Popa 1986,Remark 4.1.6] motivates the following.

Definition 3.1. Suppose that N is a II1 factor, and let N0 ⊂ N be a weakly dense∗-subalgebra of N with 1 ∈ N0. Then N0 contains a non-0 set if there is a finiteF ⊂ N0 and a K > 0 such that for all ξ ∈ L2(N , τ ), if 〈ξ, 1〉 = 0 then we have‖ξ‖22 ≤ K

∑x∈F‖xξ − ξ x‖22.

Note that by [Connes 1976, Lemma 2.4] one can check that N0 has a non-0 setif and only if there exists a finitely generated subgroup G ⊂ Int C∗(N0) such thatthere is no nonnormal G-invariant state on N . Also it follows from the definitionthat N ⊂ N contains a non-0 set if and only if N does not have property 0 of[Murray and von Neumann 1943]. Also, if 3 is a countable ICC group, then by[Effros 1975] 3 is not inner amenable if and only if C3 contains a non-0 set.

We now come to the main result, which is to give several equivalent character-izations of property (T); in particular we obtain a 1-cohomology characterizationof property (T), which is the analogue of the Delorme–Guichardet theorem fromgroup theory.

Theorem 3.2. Suppose that N is a separable finite factor with normal faithfultrace τ . Let N0 ⊂ N be a weakly dense ∗-subalgebra such that 1 ∈ N0 and N0 iscountably generated as a vector space. Consider the following conditions:

(a) N has property (T).

(b) There exists a finite F ⊂ N0 and a K > 0 such that if H is a Hilbert N-Nbimodule with ξ ∈ H and if δξ = maxx∈F {‖xξ − ξ x‖}, then there exists aξ0 ∈H such that xξ0 = ξ0x for all x ∈ N and ‖ξ0− ξ‖ ≤ δξK .

(c) Every densely defined closable derivation on N0 is inner.

(d) Every closable, conservative, symmetric c.c.n. map on N0 is bounded in ‖ · ‖1on (N0)1.

(e) There exists a finite F ′ ⊂ N0 and a K ′ > 0 such that if φ : N → N is a c.p.map with φ(1)≤ 1, τ ◦φ≤ τ , and φ=φ∗, and if δ′φ =maxx∈F ′{‖x−φ(x)‖2},then τ((y−φ(y))y∗)≤ K ′δ′φ for all y ∈ (N )1.

Then (b)⇒ (c)⇒ (d)⇒ (e)⇒ (a). If moreover N0 contains a non-0 set, thenalso (a)⇒ (b).

194 JESSE PETERSON

Proof. (b) implies (c). Let δ : N0 → H be a closable derivation, and note thatby Section 1.2, we may assume that δ is real. Let φt : N → N be the semigroupof normal symmetric c.p. maps associated with δ. Then for all y ∈ N0, we have‖δ(y)‖2 = limt→∞ τ(

1t (y−φt(y))y∗).

Let (Ht , ξt) be the pointed correspondence obtained from φt . Then since φt isunital and symmetric, ‖yξt−ξt y‖22= 2τ((y−φt(y))y∗) for all y ∈ N . Let F ⊂ N0

and K > 0 be as in (b), and let C = sup0<t≤1,x∈F τ((x −φt(x))x∗)/t . Then

‖δ(y)‖2 = limt→0

τ(1t (y−φt(y))y∗)

= limt→0‖yξt − ξt y‖22/2t

≤ 2 sup0<t≤1x∈F

‖xξt − ξt x‖22K 2/t = 4 sup0<t≤1x∈F

τ((x −φt(x))x∗)K 2/t = 4C K 2

for all y ∈ (N0)1. Thus δ is inner by Theorem 2.2.

(c) implies (d). This follows from Theorem 2.2.

(d) implies (e). Let {xn}n∈N be a sequence in (N0)1 such that N0 = sp{xn}n∈N. If(e) does not hold, then for each k ∈ N there exists a c.p. map φk : N → N suchthat φk(1) ≤ 1, τ ◦ φk ≤ τ , and φk = φ

∗

k , and there exists a yk ∈ (N0)1 such thatτ((yk −φk(yk))y∗k ) > 4kδ′k , where δ′k =max j≤k{‖x j −φk(x j )‖2}.

Let 9k = (id−φk)/δ′

k , and let 9 =∑∞

k=1 2−k9k . Then since N0 = sp{xn}n∈N,9 : N0→ L2(N , τ ) is a well-defined symmetric c.c.n. map with 9(1) ≥ 0. Alsosince φk(1)≤ 1, τ ◦φk ≤ τ , and φk = φ

∗

k , if we let (Hk, ξk) be the pointed HilbertN -N bimodule corresponding to φk then, for all x ∈ N ,

2τ((x −φk(x))x∗)≥ τ ◦φk(x∗x)+ τ(x∗xφk(1))− 2τ(φ(x)x∗)

= ‖xξk − ξk x‖2 ≥ 0.

Thus

‖9(yk)‖1 ≥ τ(9(yk)y∗k )≥ 2−kτ((yk −φk(yk)y∗k )/δ′

k > 2k for all k ∈ N.

Hence if we let 9 ′(x)= 9(x)− x9(1)/2−9(1)x/2, then 9 ′ is a closable, con-servative, symmetric c.c.n. map that is unbounded in ‖ · ‖1 on (N0)1.

(e) implies (a). Let F ′ and K ′ be as in (e), and let ε > 0. Suppose φ : N → N is ac.p. map such that φ(1) ≤ 1, τ ◦ φ ≤ τ , φ = φ∗, and ‖x − φ(x)‖2 < ε2/2K ′ forall x ∈ F ′. Let (Hφ, ξφ) be the pointed Hilbert N -N bimodule associated with φ.Then since ‖φ(1)‖2 ≤ 1 by [Popa 2006, Lemma 1.1.3], we have

‖y−φ(y)‖22≤‖yξφ−ξφ y‖2= 2τ((y−φ(y))y∗)≤ 2K ′δ′φ <ε2 for all y ∈ (N )1.

Hence N has property (T) by [Peterson and Popa 2005, Lemma 3].


(a) implies (b). If N0 contains a non-0 set, then [Popa 1986, Remark 4.1.6] showsthat [Connes and Jones 1985, Proposition 1] applies to give the desired result. �

The proof in Theorem 3.2 that (d) implies (e) can be suitably adapted to thecase of inclusions of σ -compact and locally compact groups, thus showing thatan inclusion of groups has relative property (T) if and only if “δ depends linearlyon ε”, answering a question of Jolissaint — see [2005, Theorem 1.2].

Let B ⊂ N with 1 ∈ B be a ∗-subalgebra, and suppose X = X ∈ N . Recallfrom [Voiculescu 1998] that a dual operator to (X ;B) in L2(N , τ ) is an operatorY ∈B(L2(N , τ )) such that

[B, Y ] = 0 and [X, Y ] = P1,

where P1 is the orthogonal projection onto C1.

Corollary 3.3. Suppose N is a separable finite factor with property (T), let B⊂Mwith 1 ∈ B be a ∗-subalgebra, and let X = X∗ ∈ N such that B[X ] generates N asa von Neumann algebra. Suppose that B is diffuse and B[X ] contains a non-0set. Then the conjugate variable J (X :B) does not exist in L2(N , τ ), that is,8∗(X :B)=∞. Also (X ;B) does not have a dual operator in L2(N , τ ).

Proof. If the conjugate variables J (X :B) did exist in L2(N , τ ), then we wouldhave, as in Section 1.6.1, a closable derivation on B[X ] that is not inner. Thereforeby Theorem 3.2 this cannot happen.

The fact that (X ;B) does not have a dual operator in L2(N , τ ) then followsdirectly from [Voiculescu 1998]. �

4. Property (T) and amalgamated free products

We include here an application of the above ideas, showing that a large class ofamalgamated free products do not have property (T). We first prove that if N hasproperty (T), then even though a c.c.n. maps may be unbounded on some domains,it must still satisfy a certain condition on its rate of growth.

Theorem 4.1. Suppose N is a finite von Neumann algebra with normal faithfultracial state τ . If N has property (T) and 9 : D9 → L2(N , τ ) ⊂ L1(N , τ ) is aconservative, symmetric c.c.n. map, then any sequence {xn}n in (D9)1 such that‖9(xn)‖2→∞ satisfies ‖9(xn)‖2/‖9(xn)‖→ 0.

Proof. Let {8t }t be the semigroup of unital normal symmetric c.p. maps associatedwith 9 as in Section 1.7, and for each β > 0, let εβ = supt≤β,x∈N1

‖8t(x)− x‖2.Since N has property (T), we know εβ→ 0 as β→ 0.

196 JESSE PETERSON

For all β > 0 and x ∈ (D9)1, we have∫ β

08t ◦9(x)dt = lim

s→0

∫ β

08t((8s(x)− x)/s)dt

= lims→0

1s

(∫ β

08t+s(x)dt −

∫ β

08t(x)dt

)= lim

s→0

1s

(∫ β+s

s8t(x)dt −

∫ β

08t(x)dt

)= lim

s→0

1s

( ∫ β+s

β

8t(x)dt −∫ s

08t(x)dt

)=8β(x)− x .

Hence for all x ∈ (D9)1,

‖9(x)‖2 ≤∥∥∥ 1β

∫ β

08t ◦9(x)dt

∥∥∥2+

∥∥∥ 1β

∫ β

0(8t ◦9(x)−9(x))dt

∥∥∥2

≤εβ

β+

1β

∫ β

0

∥∥(8t ◦9(x)−9(x))∥∥

2dt

≤εβ

β+‖9(x)‖ 1

β

∫ β

0εt dt ≤

εβ

β+‖9(x)‖εβ .

Thus εβ ≥ ‖9(x)‖2β/(1+‖9(x)‖β), and since εβ→ 0 the result follows. �

Corollary 4.2. Let N1 and N2 be finite von Neumann algebras with normal faithfultracial states τ1 and τ2, respectively, and suppose B is a common von Neumannsubalgebra such that τ1|B = τ2|B . Suppose also that there are unitaries ui ∈U(Ni )

such that EB(ui )= 0 for i = 1, 2. Then M = N1 ∗B N2 does not have property (T).

Proof. Let τ = τ1 ∗B τ2 be the trace for M , and let H = L2(M, τ )⊗B L2(M, τ ).Define δ to be the unique derivation from the algebraic amalgamated free productto H that satisfies δ(a)= a⊗B 1−1⊗B a for all a ∈ N1, and δ(b)= 0 for all b∈ N2.By [Nica et al. 2002, Corollary 5.4], δ∗(1⊗B 1) = 0. In particular and just as inthe nonamalgamated case, δ is a closable derivation. Furthermore if u1 and u2 arethe unitaries as above and z ∈ N0, then 〈δ((u1u2)

n), δ(z)〉 is equal to

n−1∑j=0

〈(u1u2)j u1⊗B u2(u1u2)

n− j−1− (u1u2)

j⊗B (u1u2)

n− j , δ(z)〉

=

n−1∑j=0

〈1⊗B 1, u∗1(u∗

2u∗1)jδ(z)(u∗2u∗1)

n− j−1u∗2− (u∗

2u∗1)jδ(z)(u∗2u∗1)

n− j〉,


also, for each 0≤ j < n, we may use the Leibniz rule for the derivation to rewrite(u∗2u∗1)

jδ(z)(u∗2u∗1)n− j as the sum

(3)

δ((u∗2u∗1)j z(u∗2u∗1)

n− j )+

−

j−1∑k=0

(u∗2u∗1)ku∗2δ(u

∗

1)(u∗

2u∗1)j−k−1z(u∗2u∗1)

n− j+

−

n− j−1∑i=0

(u∗2u∗1)j z(u∗2u∗1)

i u∗2δ(u∗

1)(u∗

2u∗1)n− j−i−1.

However when we take the inner product with 1⊗B 1, the first term will be 0 asmentioned above, and by freeness (since u1 and u2 have expectation 0) the otherterms will be 0, except, when i = n− j−1 in the third term. In that case, we have

−〈1⊗B 1, (u∗2u∗1)j z(u∗2u∗1)

n− j−1u∗2δ(u∗

1)〉 = −〈1⊗B 1, (u∗2u∗1)j z(u∗2u∗1)

n− j⊗B 1〉

= −τ(EB((u1u2)n− j z∗(u1u2)

j )).

Similarly,

〈1⊗B 1, u∗1(u∗

2u∗1)jδ(z)(u∗2u∗1)

n− j−1u∗2〉 = τ(EB(u2(u1u2)n− j−1z∗(u1u2)

j u1)).

Hence from the above equalities we get

〈δ((u1u2)n), δ(z)〉 =

n−1∑j=0

τ(u2(u1u2)n− j−1z∗(u1u2)

j u1+ (u1u2)n− j z∗(u1u2)

j )

= 2nτ((u1u2)nz∗).

In particular, the c.c.n. map 9 associated with δ has 9((u1u2)n) = 2n(u1u2)

n ,and so ‖9((u1u2)

n)‖2 → ∞ but ‖9((u1u2)n)‖2/‖9((u1u2)

n)‖ 6→ 0; hence Mdoes not have property (T) by Theorem 4.1. �

Note that we only used the fact that u1 and u2 are unitaries to insure that2n‖(u1u2)

n‖2 →∞. Also note that the conditions of Corollary 4.2 are satisfied

when M is a free product (with amalgamation over C) as well as when M is agroup von Neumann algebra coming from an amalgamated free product of groups.We also mention that from the calculation above we are able to compute explic-itly the semigroup of c.p. maps that δ generates — it is the semigroup given byφt = (e−2t id+ (1− e−2t)EB) ∗B id.

Acknowledgments

I would like to thank Professor Sorin Popa for his encouragement and many stim-ulating discussions. I would also like to thank Professor Dima Shlyakhtenko for

198 JESSE PETERSON

showing me to [Sauvageot 1989; 1990], and Professor Jean-Luc Sauvageot himselffor his useful comments on an earlier version of this paper.

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JESSE PETERSON

VANDERBILT UNIVERSITY

MATHEMATICS DEPARTMENT

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[email protected]

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