Available at: http://www.ictp.trieste.it/~pub-off IC/98/80

United Nations Educational Scientific and Cultural Organizationand

International Atomic Energy Agency

THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

COHOMOLOGIES AND DEFORMATIONSOF RIGHT-SYMMETRIC ALGEBRAS

(Dedicated to A.I. Kostrikin on the occasion of his 70-th birthday)

Askar Dzhumadil’daev1

Institute of Mathematics, Pushkin str.125, Almaty, 480021, Kazakhstanand

The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy.

Abstract

An algebra A with identity (a o b) o c — a o (b o c) = (a o c) o b — a o (c o b), is called right-symmetric. Cohomology and deformation theory for right-symmetric algebras are developed.Cohomologies of gln and half-Witt algebras Wnr

sym, p = 0, Wnrsym(m), p > 0, are calculated.In particular, one right-symmetric central extension of W1rsym is constructed.

MIRAMARE - TRIESTE

July 1998

1E-mail: askar@itpm.sci.kz

CONTENTS

1. INTRODUCTION.

An algebra A over a field K of characteristic p > 0 is called right-symmetric [?], [?], if forany a,b,c G A, the following condition takes place

a o (b o c) — (a o b) o c = a o (c o b) — (a o c) o b.

Any associative algebra is right-symmetric. For example, gln under usual multiplication ofmatrices is right-symmetric. An algebra of vector fields K[[x± 1,... , xn±1]] under multiplicationudi o vdj = vdj{u)di gives us a less trivial example of right-symmetric algebras. It is notassociative. Since its Lie algebra is isomorphic to Witt algebra Wn, we call it a half-Wittalgebra and denote it as Wnr

sym. If n = 1, this algebra satisfies one more identity

ao (bo c) = bo (ao c).

Such algebras are called Novikov [?], [?]. The generalisation of Novikov structure for the casen > 1 is possible, if we consider half-Witt algebra not with one, but with two multiplications. Ifwe endow K[[x ± 1 , . . . , x±1]] with a second multiplication udi * vdj = di(u)vdj, then we obtainalgebra with the following identities

a o (b o c) — (a o b) o c — a o (c o b) + (a o c) o b = 0,

a* (b* c) — b* (a* c) = 0 ,

a o (b * c) — b * (a o c) = 0 ,

( a * b — b * a — a o b + b o a ) * c = 0,

(a o b — b o a) * c + a * (c o b) — (a * c) o b — b * (c o a) + (b * c) o a = 0.

These two multiplications are useful in the construction of right-symmetric, Chevalley-Eilenbergand Leibniz cocycles of such algebras.

We develop cohomology theory for right-symmetric algebras. We endow right-symmetriccochain complex C*sym(A,M) = ®kC*sym(A,M), where Crsymk+1(A,M) = Hom(A(g>A

k(A),M) ,k > 0, by a pre-simplicial structure. Corresponding cohomologies can be "almost" obtained byderived functor formalism. The exact meaning of the word "almost" can be found in section??. Roughly speaking, this means that one should be more careful in considerating small degreecohomologies. If we take Crsym0(A, M) as M, then we should consider the operator drsymwith cubic condition d3rsym = 0. [?]. We prefer taking Crsym0(A, M) as a Kerd2rsym on M,i.e., Crsym0(A,M) := M

l.ass := {m G M : (m,a,b) = 0,Va,6 G A}. Then for any m G M, wecan correspond 2-right-symmetric cocycles, V(m) : (a, b) i-> (m,a,b). We call such cocycles asstandard. If m e Ml.ass, then V(m) = 0. If m e M, then the cohomological class [V(m)] = 0,because of V(m) = dω, where ω(a) = [a, m]. Moreover it is true, if M is a submodule of someright-symmetric A— module M, and m G M, such that [a,m] = a o m — m o a G M,Ma G M.If m G M, such that drsymfh(a) ^ M, then V(m) can give a nontrivial class of 2-right-symmetric cocycles in H% m(A,M). For example, Osborn 2-right-symmetric cocycles for A =W1rsym(m),p > 0, that appear in constructing simple Novikov algebras,

(ud, vd) i—> xp ~luvd,

(ud, vd) i—> xp ~2uvd,

are V(a; p m + 19), and V(xpmd), correspondingly.If k > 0, right-symmetric cohomologies Hrsymk+1(A, M) are isomorphic to Chevalley-Eilenberg

cohomologies Hliek(A, C1(A,M)), where Ahe— module structure on C 1(A,M) is given by a

special way: [a, f](b) = —drsymf(b,a). We endow also right-symmetric universal envelopingalgebra by a Hopf algebraic structure. It allows us to consider cup products, that are very usefulin cocycle constructions.

Second cohomology space Hrsym2(A, A) is interpreted as a space of right-symmetric deforma-tions. We calculate right-symmetric cohomologies of matrix algebra gln,p = 0. We prove that,in the category of irreducible antisymmetric glnSym— modules, nontrivial cohomologies appear

only in the case of M = (gln)anti. Moreover, right-symmetric cohomology of glnrsym in (gln)anti

can be reduced to Chevalley-Eilenberg cohomology of Lie algebra gln with coefficients in trivialmodule:

Hrsymk+1(gln, (gln)anti) = H^l^K), k > 0.

In particular, Hrsymk+1(gln, K) = 0, k > 0. We calculate also right-symmetric cohomologies of glnwith coefficients in regular module. These results show that gln has (n

2 — 1)—parametricalnontrivial right-symmetric deformations. Any formal right-symmetric deformation of gln isequivalent to the deformations given by the rule

(a,b)^aob + ttrb[X,a], X e sl2.

One can choose the prolongation in another way:

{a,b)^ aob + tX o ((tr a)b -tr{aob) + (tr b)a)

+t2{tratrb- (trao b)2X2 - (tratr(X o b))X - (tr (a o X)tr b)X} + ••• .

We prove that right-symmetric cohomologies of A = Wnrsym with coefficients in antisymmetric

modules can also be reduced to Chevalley-Eilenberg cohomologies of the Lie algebra Wn. Asit turned out, Hrsym2(A,A) for A = Wn,p = 0, or A = Wn(m),p > 0, is too large and thishappens mainly because of largeness of a space of right-symmetric derivations. There is animbedding

Zrsym1(A, A) HUA U) -+ Hr

sym2(A, A).

We prove that Zrsym1(A, A) has a basis consisting of two types of right-symmetric deriva-

tions: di,i = 1,... ,n, if p > 0, one should consider also derivations

0) E?=i E ^ l V hkMh» s u c h t h a t didjius) = 0, i, j , s = 1,... , n, \ M e K. We formulate a re-sult about local deformations of Wn,p = 0, or Wn(m),p > 3. The space H

r

sym2(Wn, Wn),p = 0,is generated by classes of cocycles of four types. In the case of p > 3 Steenrod Squares alsoappears. We prove that W1rsym has exactly one right-symmetric central extension. It can begiven by cocycle

-i, p = 0,

_l, p>0.

For n>1, Hrsym2(Wn,K) = 0.For right-symmetric algebras, Novikov algebras and some cohomology calculations see also

[71 [71 [71 [71 [71 [71 [71

2. RIGHT-SYMMETRIC ALGEBRAS AND (CO)MODULES.

2.1. Right-symmetric algebras. An algebra A over a field K with multiplication (a, b) i—>a o b, is called Lie-admissible, if the vector space A under commutator [a,b] = a o b — b o a canbe endowed by a structure of Lie algebra. An algebra A is right-symmetric, if it satisfies thefollowing identity :

(a o b) o c — a o (b o c) = (a o c) o b — a o (c o b), Va, b,c e A.

Let (a, b, c) = a o (b o c) — (a o b) o c be the asssociator of elements a,b,c e A. In terms ofassociators the right-symmetric identity is

(a, b, c) = (a, c, b), Va, b, c G A.

Right-symmetric algebra A is Lie-admissible. Similarly, one can define left-symmetric algebraby identity

(a, b, c) = (b, a, c), Va, b,c e A.

Categories of left-symmetric algebras and right-symmetric algebras are equivalent. Any left(right)-symmetric algebra will be right(left)-symmetric under new multiplication (a, b) i—> b o a.

An element e of right-symmetric algebra is called left unit, if e o a = a, for any a G A.Denote by Ql(A) a space of left units. Let Zl(A) = {z £ A : z o a = 0,Va £ A} be left center ofA. Call a space Nl(A) = Zl(A) ®Qi{A) as a semi-senter of A. Then [Nl(A), Nl(A)] C Zl(A).An algebra A is called (left) unital, if it has nontrivial left units.

Any associative algebra is a right-symmetric algebra. In such cases, we will use notations likeAass, if we consider A as associative algebra and Arsym, if we consider A as right-symmetricalgebra. Similarly, for right-symmetric algebra A notation Arsym means that we use onlyright-symmetric structure on A and Alie stands for a Lie algebra structure under commutator

Matrix algebras gln gives us examples of unital right-symmetric algebras.Less trivial examples appear in the consideration of Witt algebras. The algebra Wn,p = 0,

and Wn(m) defined below has not only right-symmetric mutiplication (a, b) i-> a o b, but alsoone more multiplication (a, b) i—> a * b, that satsifies the following identities

a o (b o c) — (a o b) o c — a o (c o b) + (a o c) o b = 0,

a* (b* c) — b* (a* c) = 0 ,

a o (b * c) — b * (a o c) = 0 ,

( a * b — b * a — a o b + b o a ) * c = 0,

(a o b — b o a) * c + a * (c o b) — (a * c) o b — b * (c o a) + (b * c) o a = 0.

Letk

i=1

be an algebra of Laurent power series, if the main field k has characteristic 0 and

U = O n ( m ) = {x(α) =l[x(

i

ai) : α = ( α 1 , . . . ,αn),0 < α i < pmi,i = 1,... ,n}

be a divided power algebra if char k = p > 0. Recall that On(m) is pm -dimensional and the

muliplication is given by

where m = J2i^iy a n d

a JLet €i = (0 , . . . , 1 , . . . ,0). Define di as a derivation of U,

i

dt{xa) = a%x

a~^, p = 0,

9i(a; ( a ))=a; ( a- e i ) , p > 0.

Endow a space of derivations Der U = {J2i Uidi : ui G U} by multiplications:

u9i o vdj = vdj(u)di,

udi * vdj = di(u)vdj.

Denote obtained algebra as Wnrsym(m). If p = 0, this denotion will be reduced until Wnrsym

nIf n ^ 0(modp), then an element e = J2ixidi/n is a left unit of Wnrsym(m). If n = 1, then

aob = a*b. Thus the algebra A = W1rsym(m) in addition to right-symmetry condition satisfies

the following identitya o (b o c) = bo (a o c), Va, 6 , c e A

Such algebras are called Novikov algebras [?]. Notice that Novikov algebra W1(m) is unital. Ifright-symmetric algebra A is Novikov algebra, we will use denotion Anov.

2.2. Right-symmetric modules and comodules. A vector space M is said to be moduleover right-symmetric algebra A, if it is endowed by right action

M x A^ M, (m,a)\-^moa

and left actionA x M ^ M, (a, m) i—> a o m,

such that

m o [a, b] — (m o a) o b + (m o b) o a = 0,

(a o m) o b — a o (m o b) — (a o b) o m + a o (b o m) = 0,for any a,b G A,m G M. We will say, that M is antisymmetric A— module, if the left action ofA is trivial, i.e., aom = 0, for any a G A,m G M. For module M over right-symetric algebraA, denote by Manti its antisymmetric A— module: Manti = M, (m, a) i—> m o a, (a, m) i—> 0,for all m G Manti, a e A.

A right-symmetric A— module M is said to be special, if the right action satisfies the followingcondition

m o (a o b) - (m o a) o b = 0, Va, b G A, Vm G M.

A special module is antisymmetric, if a o m = 0, for all a e iExample. For right-symmetric algebra A its vector space A can be endowed by a natural

structure of A— module, (a, m) i-> a o m, (m, a) i-> m o a, a, m G A. In such cases we say thatM = A is regular A— module. If A is associative algebra, then regular module is special.

The functor

Alie-module —> right Alie-module —> Antisymmetric A-module

gives us an equivalence of the category of antisymmetric A— modules to the category of (right)Ahe— modules. Antisymmetric A -module corresponding to right Alie -module M will bedenoted by Manti.

Assume that A is an associative algebra A with multiplication (a, b) i—>• a • b. In the last caseof Arsym right-symmetric multiplications can be defined in two ways: by (a, b) i—> a • b or by(a, b) i—b 6 • a. For definiteness we endow Arsym by multiplication (a, b) i—*• a • b. For associativealgebra A the functor

Right Aass-module —• Antisymmetric special Arsym-module

gives us an equivalence of the categories of antisymmetric special Arsym -modules and rightAass -modules.

Right-symmetric A -module M can be endowed by a structure of module over Lie algebraAlie by action [a,m] = a o m — m o a. The obtained module is denoted by Mlie.

So, for defining module structure on a vector space M over a right-symmetric algebra Aone should define on M right module structure over the Lie algebra Alie and endow it by aleft action that satisfies condition (AAM). As we mentioned before the last can be done by atrivial way by setting a o m = 0, Va G A, Vm G M.

For a module M over right-symmetric algebra A the subspaceMl.ass = | m . e M . ^ (h Q = Q) Vfflj ft e Ay

is called a left associative invariant subspace of M, and

Ml.inv = {m G M : m o a = 0, Va G A}

is called a left invariant subspace of M. If M = A is regular module, then Al . a s s is called a left

associative center. Notice that, Al.inv coincides with the left center of A. Notice that, Ml.inv

is close under right action of A andl.inv (— l.ass

Module M of (left) unital right-symmetric algebra is called (left) unital, if

e o m = m, Ve G Qi{A),\/m G M.

and (left) central, ifzom = 0, ,Vze

Regular module of unital right-symmetric algebra is unital and central.A vector space M is called comodule over right-symmetric algebra A, if there are given right

actionM x A—> M, (m,a) \-^ mo a,

and left action

Ax M —• M, (a, m)^aom,such that

[a, b] o m — a o (b o m) + b o (a o m) = 0,

—b o (m o a) + (b o m) o a — m o (a o b) + (m o a) o b = 0,

for any fl,f)6A,meM. A comodule M is special, if it satisfies the identity(a o b) o m = a o (b o m), Va, b e A, Vm e M.

A (special) comodule M is called antisymmetric, if m o a = 0, for any a £ A.Example. Let A be right-symmetric algebra, M be A—module and M' = {f : M —K }

be a space of linear functions on M. Set

(a o f)(m) = f(m o a), (f o a)(m) = f(a o m).

Then M' under actions (a, f) i-> a o f, (f, a) i-> f o a, can be endowed by a structure ofA—comodule. Check it.

{[a,b]f-ao(bof)+bo(aof)}(m) =

f(m o [a, b] — (m o a) o b + (m o b) o a) = 0,

{-b o (f o a) + (b o f) o a - f o (a o b) + (f o a) o b}(m) =

/(—a o (m o b) + (a o m) o b — (a o b) o m + a o (b o m)) = 0.

The A— comodule A' for regular module A is called coregular comodule of A. If A is asso-ciative, then A' is special comodule.

For A— comodule M let

Mr.ass = {m G M : (a, b, m) = 0, Va, b e A}

be a right associative invariant subspace of M and

Mr.inv = {m G M : a o m = 0, Va e A}

be a right invariant subspace of M. Notice that Mr.inv is close under left action of A. Aninclusion takes place

r.inv f— r.ass

2.3. Antisymmetric module Cright1(A,M).

Proposition 2.1. The space of linear maps Crsym1(A,M) := C1(A,M) = {f : A—> M} can be

endowed by a structure of antisymmetric A— module, where the right action is given by

(foa)(b)=f(b)oa-f(aob)+bof(a), a,b e A.

Proof. For f e C1(A, M), a,b,c

+ao[f,c](b)-[f,c](aob) + [f,c](a)ob =

ao f([b,c]) - f(ao[b,c]) + f(a)o[b,c]

-a o drsymf(c, b) + drsymf(a o c, b) - drsymf(a, b) o c+

+a o drsymf(b, c) - drsymf(a o b, c) + drsymf(a, c)ob =

ao f([b,c]) - f(ao[b,c}) + f(a) o[b,c]

— ao (co f(b)) + a o f (co b) — ao (f(c) o b)

+ (a o c) o f(b) — f((a o c) o b) + f(a o c) o b

— ao f(b) o c + f (ao b) o c — (f(a) o b) o c +

+ a o (b o f(c)) - a o f(b o c) + a o (f(b) o c)

— (a o b) o f(c) + f((a o b) o c) — f(a o b) o c +

(a o f(c)) o b — f(a o c) o b + (f(a) o c) o b =

= 0.

So, C1(A,M) is a right Alie -module. •

2.4. Universal enveloping algebras of right-symmetric algebras. Consider two copies ofA, denote them by Ar,Al, and the tensor algebra T(Ar © Al). Algebras Ar,Al supposed tobe free as /C— module and the tensor algebra T(Ar © Al) is associative and unital. Elementsof Ar and Al corresponding to a G A denote as ra and la. Let U(A) be a factor-algebra ofT(Ar®Al over an ideal J generated by r[a,b — rarfe + rbra, [rb,la] — hla + laob- This algebra canbe considered as a universal enveloping algebra of right-symmetric algebra A. Denote by U{A)the factor-algebra of T(Ar (BA1) over an ideal J generated by {raob — rarb, [rb,la] — hla + laob}-This algebra is called a special universal enveloping algebra of A. Notice that, J C J, since

r[a,b] ~ Va, n] = {raob - rarb} - {rboa + rbra} e J.

So, the following exact sequences of algebras take place

0 ->• J ->• T ( A r © A l ) -»• ( A ) -»• 0,

0 ->• J ->• T ( A r © A l ) -»• f / ( A ) ->• 0,

and0 -»• J/J -»• U ( A ) -»• f / ( A ) - > 0.

In particular, we can consider C/(A) as right U(A)— module:

where u and -u are elements of U{A) and U(A) corresponding to u

Theorem 2.2. Let A bea right-symmetric algebra.i) There exists an equivalence of the categories of A— modules and right U(A)— modules. The

same is true for A— comodules and left U(A)— modules.ii) The category of special A— modules is equivalent to the category of right U{A)— modules.

The same is true for special A— comodules and left U{A)— modules.

Proof. i) Let (r, l) : A —E nd M be a representation of right-symmetric algebra A corre-sponding to A -module M, i.e.,

r : A —> EndM, a i-> ra, mr a = mo a,

l : A —> EndM, a i-> la, mla = a o m,

linear operators, such that for any a, b G A,

K rb,la] " lbla + U& = 0.So, any A— module is a right U(A) — module and, converse, any right U(A) — module can beconsidered as an A— module.

A corepresentation (rco, lco) : A —>• End M, corresponding to A-comodule M,

rco : A^ EndM, a >->• ra, racom = a o m,

lco : A -> EndM, a ^ /a, ! > = m o a,

satisfies conditions (MAA), (AAM) for raco,1™• So, any A— comodule is a left U(A)— module.Any left U(A)— module can be considered as a A— comodule.

ii) Let M be a special A— module. Then by the rule

mra = mo a, mla = m o a,

we obtain a right U{A)— module:

m o ( a o f ) ) - ( m o a ) o f ) = 0-» rao& = rarb.

Converse, for a right U(A)— module N , one can correspond special A— module N, by no a :=nra,a on = nla.

For a special A— comodule M notice that

(ooft) om -ao (bom) = 0 => raco•b = racorb

co,

if rac°m = aom,Vfm = m o a. So, any special A— comodule is a left Uspec(A)— module. Aconverse statement is also evident. •

2.5. Right-symmetric cohomologies as a derived functor . Recall that factor-imagesof the element u G T(Ar © Al) in U(A) and U(A) are denoted by u and u. ConsiderA = A(& < 1 > as a right U{A)— module:

1 o fa = ra o 1 = a , 1 o la = la o 1 = a , a o fb = a o b, a o l b = b o a ,

for all a G A. Endow U(A) by a structure of U(A)— module as in the subsection ??. Con-sider A k(A) (A),k > 0, as a right U{A)- module. Then A ® f\kA ® U(A) is a freeU(A)— module. Denote its generators by < a0, a1,... , ak >, where a0 G A, a 1 • • • a k G AkA.Construct homomorphisms

d : A eg) AfcA eg) f/(A) -»• A eg) Ak~lA eg) f/(A), A; > 0,

d : A eg)

e : U(a) ->• A ,

as below =

1 ) i + 1 < a i, a 1 , . . . , a &i,... , ak > laoi=1

+ ( - 1 ) * < a0 o ai, a 1 , . . . , a di}... , ak >

+ ( - l ) i + 1 < a0, a 1 . . . ,di,... ,ak> fai}

i + 1 < a 0 , a 1 , . . . , d i , . . . , a j - i , [ a i , a j ] , . . . , a k

10

lao- < 1 > rao,

el = 1, era = a, ela = -a.

Then the following sequence

• • • —> A A2A U(A) —> A A C/(A) —> A £

is almost a free resolution of the right [/(.A)— module A. Here the words "almost free" meanthat all members of the resolution except U(A) is are free right U{A)— modules.

Notice thatHomU(A)(A 0,

and Homu(A)(U(A),M) consists of g : U(A) —> M, such that

J l i^J i^f l^ 'aob) g\^-\'a'b 'aobj)

n i iy* IY*-I 'Y* i \ — ft i 'Y* 'Y*! 'Y* i \ —

y\' a1 b I aob) —ij\> a> b > aob) —

0 ,

for any a, b G A. So,

Homu{A)(U(A),M) ^ {m G M : (m,a,b) = 0}.

Therefore, as a right-symmetric cochain complex we can take

Crsym0(A, M) = {meM : (m, a, b) = 0, Va, b e A},

Ck+lm(A,M)=A®AkA, k>0.

These statements will follow from our results on right-symmetric cohomologies in the next sec-tions. Our approach is slightly different from Koszul's approach. We will argue in cohomologicalterms and prove that right-symmetric cochain complex has a pre-simplicial structure.

Let us mention these results relating homologies. Let M be comodule over right-symmetricalgebra A. Endow M A by a structure of antisymmetric A— comodule with a left action

5 o (m ® a) = mo a®b — m®f lo6 + ( i o m ® a .

SetC0rsym(A, M) := Mr.ass :={meM : (a, b, m) = 0, Va, b G A},

Crks^(A,M)=M(S)A(S>Ak(A), k>0.

Clsym{A,M) = ®kCrksym{A,M).

Then Clsym(A,M) is chain complex under the boundary operator

8 : Ckrsym+1(A, M) -* Ckrsym(A, M ) ,

d(m ao a,\ A • • • A ak) =

k

J 3 ( — l ) t + l { m o ao (8) a% ® a\ A • • • di • • • A aki=1

—m (g> ao o a,i (g> a\ A di • • • A at

+(ii o m ao a,\ A • • • di • • • A ak }

iy+1m

11

rsymco(x)(m a0 a1 A • • • A ak) =

k

^ ( — l ) * ( m o ao ® (M ® a\ A • • • di • • • A cik

i=1

—m ® a0 o ai ® a1 A ••• di ••• A ak

+ai o m ® a a1 A • • a di • • • A ak}

— i y + 1 m (g) ao a i A • • • A a j _ i A [x, a,i] A • • • A ak,

and an isomorphism of A— comodules takes place

Ckrsym+1(A, M) ^ Cklie(A, M®A).

that induces an isomorphism of homology spaces

Hkrsym+1(A, M) ^ Hkl

ie(A, M®A), k > 0.

2.6. Comultiplication of universal enveloping algebra. Let U(A) be the universal en-veloping algebra of a right-symmetric algebra A. As we noticed in section ??, it can be generatedby the elements ra,la,a G A, such that

r[a,b] ~ [ra, n] = 0, [la, n] - laob + Wa = 0, a, b G A.

Define homomorphismA:U(A) -^U(A)®U(A),

byA(l) = 1(8.1,

A(ro) = ro(8)l + l (8)( r o -Z o ) ,

A(ZO) = la eg) 1.

Since, according to right-symmetry identities,

A([ro,r6]) =

A(ra)A(r6) - A(r6)A(ra) =

r a r eg) 1 + 1 ® (ra - la)(rb - lb) - rbra ® l + l®(rb- lb)(ra - la) =

r[a,b] ® 1 + 1 ® (r[a,b] ~ l[a,b] =

(la ® 1)(rb ® 1 + 1 ® (rb - lb)) - (r eg) 1 + 1 eg) (rb - lb))(la (8) 1) - laob ® 1 + hL ® 1 =

([la,rb] - laob + lbla) ® 1 =

0.

this definition is correct.

Theorem 2.3. For a right-symmetric algebra A and its universal enveloping algebra U(A) thefollowing diagram is commutative

U(A) A U(A) ® U(A)I A | A (8)1

U(A)®U(A) ^ A U(A)®U(A)®U(A)

12

Proof.We must check that

(1 eg) A)A(u) = (A eg) l ) A ( u ) , Vu G U(A).

We have

(1 eg) A ) A ( r o ) =

( r o eg) 1 eg) 1 + 1 eg) r a eg) 1 + 1 eg) 1 eg) (ra - la)) -l(gla(gl

{ra ® 1 1 + 1 r a 1 - 1 la 1) + 1 1 {ra - la)

A ( r o ) (8) 1 + 1 (g (ra - la) =

(Acx)l)cx)A(r a ),

(1 (g A)A(la) = /oll = (A<

Similarly, homomorphism Ai defined below is also comultiplication,

Ai :U(A) -^U(A)(gU(A),

Ai(l) = 1(8)1,

Ai(r o ) = (ra - la) (g 1 + 1 eg) r a,

Ai(/O) = 1 ZoSo, we can construct for given A— modules M and N their tensor products M (g N with

a module structure unduced by comultiplication A :

(m n) o a = m o a n + m [n, a],

ao(m®n) = ao m (gn.

Moreover, it is possible for right-symmetric A— module M and for Ahe— module N. Thesemodule structures on tensor products are associative: if M, N, S are modules over right sym-metric algebra A, then

(M(gN)(gS^M(g(N(gS).

Definition. For given modules M, N over right-symmetric algebra A, a homomorphism ofA— modules M N —> S is called a cup product of M and N.

Denote the image of m (gn in S by mUn. Thus, a bilinear map

M x N —> S, (m, n) i-> m U n,

is said to be the cup product (pairing) of M and N to S, if

(f/iUrj) o a = m o a U n + m U [n, a],

a o (m U n) = a o m U n,

for any a G A,m,n G M.Let

C 1 ( A , M ) = Homk(A,M), Cliek(A,M) = Homk(/\kA,M), k>0,

(A, M) = Homk(A eg) k(A),M), k>0.

Proposition 2.4. Crsymk+1(A, M) has an antisymmetric A—module structure, where the rightaction

is defined by

(ipoX)(a0,ai,... ,ak) =

a0 o ψ(x, a1 ,.. , ak) — ψ(a0 o x,a1,... , ak)

k

) , a \ , . . . , ak) o x + ^ J V ^ O ) CL\, • • • ,di-i,[x,ai],... , a k ) ,

i=1

), k>0.

13

Proof. Since,Crsym1(A,M)=C

1(A,M),

an isomorphism of linear spaces takes place

G : C r s y m 1 ( A , M) k ( A , k) -* Crig

htk+1(A, M ) , k > 0 ,

(G(f ( g ) i p ) ) ( a o , a i , . . . , a k ) = f ( a 0 ) ψ ( a 1 , . . . , a k ) .

In section ?? we have constructed an antisymmetric right-module structure on Cright1(A, M). Liemodule structure on Cliek(A,k) over Al

ie is well known. So, for an antisymmetric A— modulestructure

Clrsym{A, M) CUA, k) = {f®:fe Cl

nght{A, M), e Cfie{A, k)}

we have

(( φ) o x)(a0 ® ( a 1 , . . . , ak)) =

( ( / o x) eg) tp)(a0 eg) ( a i , . . . , a f c)) + (/ eg) fo/>, a;])(a0 (8) ( a i , . . . , ak)) =

( f ( a 0 ) o i - f ( a 0 o x) + a 0 o f(x)) ( a 1 , . . . , a k )

a 1 , . . . , [x ,a i ] , . . . ,ak) =i=1We see that

Therefore, (ψ,x) t-^ ψ o x gives us a right representation. •

2.7. Right-symmetric modules for Wnrsym. Let

L r a = {α = ( α 1 , . . . , α n ) , α i e Z , i = l,... ,

T+ = { a e T n : a i > O , i = 1 , . . . , n } .

a n dΓn(m) = {a eT+ : αi < p

mi,i = 1,... ,n

if p > 0, m = (m1,... , mn).Let

U = K[[x±1,... . ^ l ^ ^ i a G r , , } ,

if p = 0, andU = O n ( m ) = {x(α) : α G Γ n ( m ) } ,

if p > 0.F o r p = 0, let A = Wnr

sym, if U = K[[x±1,... ,x±n1]], a n d A+ = W n + r s y m , if U+ =K[[x1,... ,xn]]. Let A be Wnrsy

m(m), if U = On(m), p > 0. Algebras A,A+ are right-symmetric and U is associative commutative.

Notice that U has a structure of antisymmetric graded A— module. The right action is givenby MO adi = adi{u). The gradings are given by

I «i \r^ n/(Y)i

C/ = ®kUk, Uk = {ueU:\u\= k } ,

A = ®kAk, A k = {adi : | a | = k + 1,i = 1,... , n } , AkoAiQ A k + l

Notice that Ao ^ glnrsym.

14

Let A0 = ®kAk, and A+0 = ®k> 0A+k, if p = 0. Let M be AQ- module, if p > 0, and— module if p = 0. Define antisymmetric A— module structure on U M 0 by (see [?])

(« m) o adi = adi{u) ® m + J ^ ucf(a) [m, a;^9j], p > 0,

(u(g>m)o adi = adi{u) ®m+ ^2 (I / P^ud13 (a) [ m , ^ ^ ] , p = 0.

3. Cohomologies of r i g h t - s y m m e t r i c a l g e b r a s

3.1. Pre-simplicial structures on C*fyl

m{A,M). For a right-symmetric algebra A and its

module M we introduce a structure of pre-simplicial cochain complex on C*fym(A, M) =

M), where

Crsy

mk+1(A, M) = Hom(A k A , M ) , k > 0.

Define linear operators A : C*+Jm(A, M) -> C*+Jm{A, M), i = 1, 2,. . . , by the rules

Di : C r s y m k(A, M) —> Crsymk+1(A,M),

Diψ(a0,a1,... ,ak) =

a0 o ψ(ai, a1,... ,di,... , ak) - ψ(a0 o f l , , o i , . . . , di}... ,ak)

0 , a 1 , . . . ,di,... ,ak) oca + ^ 0 ( a o , a i , . . . , O j , . . . , O j - i , [ O J , % • ] , . . . , a k ) ,

0 < k, i k.

Here a means that the element a is omitted.In the next section we will endow C*sym(A, M) = ®kC^sym{A, M) by a structure of cochain

complex, whereCrsymk(A,M) = 0, koC^sym(A, M)by a pre-simplicial structure:

DjDi = DiDj_1, i

15

a0(ψ(aj, a1,. .., ,di,... , [ai, as],... , ak)),

- ( a 0 o aj)ψ(ai, a1,... , a* , . . . a d j , . . . ,ak)

+ψ((a0 o aj) o ai, a 1 , . . . , a ; , . . . ,dj,... , ak)

Y ψ(a0 oa,j,ai,... r ^ — ,...,[ai,as],... ,ak),s,s^j >cococ<

X3 =

+ ( a 0 ψ ( a i , a 1 , . . . , d i , . . . ,dj}... , a k ) ) a j

- ( i p ( a 0 oa,i,ai,... , d i , . . . ,dj}... , a k ) ) a j

,di,... , as-\, K, as],... , ak))aj

j

16

Y2 =

( a 0 o a i ) ψ ( a j , ax,.., , d i , . . . ,dj}... , a k )

) o ai) o aj, a1,... ,a%,... ,a,j,... , ak)

i o ai,a1,..., ,di,.., ,dj,... ,as-i, [aj,as],... ,ak),

Y3 =

o a j , a 1 , . . . , a i ; . . . a dj,..., ak))ai

a0, a1,. .., ,di,... ,dj,... , ak)aj)ai

Present Y4 as a sum

Y4 = I V +14,2+14,3,

whereF Y^ Dj-iip(a0, a1,... ,di,... , as-i, [ai, as],... ,ak),

Y4,2 = Dj_iip(ao,ai,... ,di,... ,a,j-i, [ai,aj],... ,ak),

j_1ψ(a0, a1,... ,di,... ,as-i,[ai,as],... ,ak).

These elements can be expressed in the following ways

Y4,1 =

a j , a 1 , . . . , d i , . . . ,as-i,[ai,as],.., ,dj}... , a k ) )

• ,as-i, [ai,as],.., ,dj,... ,ak))aj

1,[ai,as1],... ,dj,... ,as-i,[aj,as},... , a k ) ,

Y4,2 =

,... ,di,... ,dj,... ,ak))

j

17

,ak)

j

18

Zrsymk(A, M) = {ψ G Cr

symk(A, M) : drsymψ = 0},

be spaces of right-symmetric cocycles,

B*rsym(A,M) = ®kBk

sym(A,M),

Brsymk(A,M) = {drsymω : ω G Ck-lm(A,M)},

be spaces of right-symmetric coboundaries, and

H*sym(A,M) = (BkHk

sym(A,M),

Hrsymk(A,M) = Zr

symk(A,M)/Br

symk(A,M),

be right-symmetry cohomology spaces.Definitions. For any x G A the interior product endomorphism i(x) of C*sym(A, M) is

defined byi(x) : Crsymk+1(A,M) -+ Crsymk(A,M),

i(x)ψ(a0,..., ak-\) = ψ(a0,x, a1,... , ak-\), k > 0,

Let ρlie : Alie —> C*fym{A, M) be a representation of Lie algebra Alie corresponding to anti-

symmetric representation

ρrsym : A -• C*fym(A, M), ρrsym(x)ψ = 0, ψρrsym(x) = ψ O x,

constructed in proposition ??,

(ρlie(x)ψ)(a0,a1,... ,ak) =

-a0 o ψ(x, a1,... , ak) + ψ(a0 ox,ai,... ,ak)

k

-ip(ao,ai,... ,ak)ox-^2tl){ao,ai,... ,ai-i,[x,a,i},... ,ak).i=1

Recall thatρlie(x)ψ = -1pplie(x) = ~1p O x = -1pprsym(x).

Proposition 3.2. (Cartan's formulas) Consider C*+lm(A,M) := 1,

(ii) i(x)D1 = -pue(x),

(iii) ρlie[x,y] = [ρlie(x),ρlie(y)],

(iv) [i(x),ρlie(y)] = -i([x,y]),

(v) drsymi(x) + i(x)drsym = -pue(x),

Proof. (i)

Dlψ(a0,x,a1,... ,ak-i) =

a0oip(ai-i,x,ai,... ,a{-i,... ,ak-\)

-ip(a0 oai_i,x,ai,... ,a{_x,... ,ak-\)

a0,x,a1,... , a f_ i , . . . ,a f c _i) o at_i

l-Kj

a 0 o i(x)ip(ai_i,ai,..., a{-X,..., a f c _ i )

19

-i(x)ip(a0 o ai-i,a\,..., a{-\,,..., ak-\)

+i(x)ψ(a0,a1,..., atl i,..., ak-i) o at_i

+ ^2 i(x)ψ(a0,a1,..., af-\,..., Oj_i, [ai, aj],... , au-i) =

l-Kj

Di_ii(x)ip(aOl... ,a f c_i).

(ii)i(x)D1ψ(a0, . . . ,a fc_i) =

D1ψ(a0,x,a1,... ,a fc_i) =

a 0 o ψ ( x , a1,... , ak-i) - ψ(a0 ox,ai,... , ak-i)

,... ,ak-i) o x + ^2i>{ao,ai,... ,aj-i,[x,aj],... , a f c _ i )

(iii) P r o p o s i t i o n ? ? .

(iv) We o b t a i n

-{(i(x)pue(y))ip}(a0, a1,... , a f c _ i ) =

(ipoy)(a0,x,ai,... , a f c _ i ) =

a 0 o (ψ(y,x,a1,... ,ak-i)-ip(a0 oy,x,ai,...

fc-i

and

) o y}(a0, a1,... , a f c _ i ) =

,a1,... ,a fc_i)} - (x

+{(i(x)ψ)(a0, a 1 , . . . , afc_i)} o y

fc-i

a0o(ip(y,x,ai,... ,ak-i))-ip(a0 oy,x,ai,

fc-i

Thus

{(-i(x)pue{y) + ρlie(y)i(x))ψ}(a0,a1,... ,afc_i)

, a 1 , . . . ,a f c_i).

20

(v) According (i) and (ii),

-Pue{x)

Thus,

3.3. Long exact cohomological sequence. The following theorem follows from standardhomological results.

Theorem 3.3. Let A by a right-symmetric algebra and

0-> M - > T -> S ->0

be a short exact sequence of right-symmetric A -modules. Then an exact sequence of right-symmetric cohomology spaces take place

0 - Zrsym0(A, M) -+ Zrsym0(A, T) -+ Zrsym0(A, S) -°>

Zlrsym{A, M) -* Zl

rsym{A, T) -* Zl

rsym{A, S) ^ H2

rsym{A, M) -* • • •

-+ H*sym(A, M) -* H*sym(A, T) -* Hk

rsym{A, S) ± H^A, M) -* • • •

Here δ is a connected homomorphism:

= [drsymφ], [ψ] S Hrsymk(A, S),k>1,

5ipl = [drsymφ1], ψ1 S Zrsymi(A, S), i = 0,1,

where φ G Zrsymk(A,T) is a representative of the cohomological class [ψ] and φ1 G Zrsymi(A,T)

moves to ψ1 under a natural homomorphism Zrsymi(A,T) —>• Zrsymi(A,S),i = 0,1.

Define a homomorphism V : Sl.ass —>• Zrsym2 (A,M), as a composition

v : Crsym0(A,S) ~^ Brsym1(A, S) —δ Zrsym2(A, M), s i—>• drsyms i—δ (drsyms).

Then

V(fh)(a, b) = fh o (a o b) — (m o a) o b.

In particular, there exist homomorphisms

5 : Sl.ass ^ Z r s y m 1 ( A , M ) , δ(s) :a»[a,s],

5 : SLinv ->• Zl (A, M ) , -5(s) :a^aos.

21

3.4. Connections between right-symmetric cohomologiesand Chevalley-Eilenberg cohomologies. Recall that for any Ahe— module Q a standardrepresentation α : Alie —> Cfie(A, Q) is given by

α(x)ψ(a1,... ,ak) =

k

[x,ψ(a1,... ,ak)] - ^ ^ ( a i , . . . , a * _ i , [x,ai],... , a k ) ,

%=\

where (x, q) i—> [x, q] is representation corresponding to the Lie module Q and ψ G Cliek(A, Q).

Theorem 3.4. Let A be a right-symmetric algebra and M be an A— module.An operator

F : Cliek(A,C1(A,M)) -+ Crsymk+1(A,M), k > 0, (2)

defined by the ruleFψ(a0,a1,... , ak) = -ip(ai,..., a f c_i)(a0)

induces an isomorphism of Ahe— modules. Moreover, F induces an isomorphism of cochaincomplexes C*fym{A,M) and C*ie(A, C

1 (A, M)). In particular,

Hrsymk+1(A, M) - Hliek(A, C1(A, M)), k>0. (3)

The following sequence is exact

0 -+ Zrsym0(A, M) -* Crsym0(A, M) -* Hlie0(A, C1(A, M)) -* Hrsym1(A, M) -* 0. (4)

Proof. Prove that for any x G A, k > 0, the following diagram is commutative

Ctte(A,CHA,M)) eM Cfte(A,C

l(A,M))IF IF

F o r ψ G Cliek(A,C1(A,M)) we have

F{α(x)ψ}(a0,a1,..., ak+1) = -g(x)ip(ai,..., ak+1)(a0) =

k

-{xo (ψ(a1,... ,ak))}(a0) +^2tp(ai,... ,a*_i, [x,ai],... ,ak)(a0) =i=1

k

a1, . . . ,ak))ox}(a0) +^V; (ai,--- ,ai-i, [x,ai],... ,ak)(a0) =

i=1

- a 0 o ψ(x, a 1 , . . . , a k ) + ψ(a0 ox,ax,... , a k ) - ψ(a0, a1,... ,ak)oxk

-^2ip(ao,ai,... ,[x,ai],... ,ak) =%=\

-(ipprsym(x))(a0, a1, . . . , ak) =

{(Fψ)ρrsym(x)}(a1,... ,ak)(a0).

Thus, F : Cliek(A, C1(A, M)) -»• Cr

symk+1(A, M) is a homomorpism of Alie- modules. It is evidentthat, F has no kernel and F is epimorphism.

Now, prove that for any k > 0 the following diagram is commutative

Cliek(A,C1(A,M)) ^

[F [F

22

For i)^Cfie{A,C\A,M)),

F(dlieψ)(a0,a1,... ,ak+1) = dlieψ(a1,... ,ak+1)(a0) =

- l ) > ( a i , . . . ,di,... ,[ai,aj],... , a k + 1 ) ( a 0 )

k+1

^ ( - l ) l [ a i , ^ ( a i , . . . , a i , . . . ,ak+1)](a0) =i=1

( - i y + 1 F t p ( a 0 , a 1 , . . . , d i , . . . , [ a i , aj],... , a k + 1 )

k+1^2(-iydrsym(ip(ai,... ,di,... , a k + 1 ) ) ( a 0 , a i) =

k+1

Fψ(a0, a 1 , . . . , a i ; . . . , [ai,a j ] , . . . .

i a 0 o (ψ(a1,... , a i ; . . . , ak +1))(ai)

k+1

k+1

>--- , a i , - - - , a k + 1 ) ( a 0 ) )

k+153(-l ) l a 0 o (Fψ(ai, a1,... , a i ;...%=\

k+1

+ 5 3 ( ~ 1 ) t i ? V ; ( a o oa,i,ai,... ,di,... ,ak+1)%=\

k+1

= drsym(Fψ)(a0, a1,... , ak+1).

Thus we obtain the equivalence of cochain complexes (Bk>oC]}e(A, C1(A, M)) and ®k>iC^sym{A,M).

In particular, an isomorphism (??) takes place.Since, Hrsym1(A,M) = Zrsym1(A,M)/Brsym1(A,M), and

Zrsym1(A, M) = {tpe C1(A, M) : drsymψ = 0} = Zlie0(A, C1(A, M)),

Brsym1(A, M) = {drsymm :m Clie(A,M) be a linear operator, such that

i=1

Introduce subspacesC* {A,M) = ®kC

k

rsym{A,M),

23

Cksym(A, M) = {tP0.

Theorem 3.5. Let A be a right-symmetric algebra and M be an A-module. Then the operatorf : C*sym(,A, M) —> C*ie(A,M) is the homomophism of cochain complexes and the folllowingcohomological sequence

H°rsym(A, M) -+ Hrsym2(A, M) -+ Hli

e2(A, M) ± Hl

rsym{A, M) -

± Hk;2m(A, M) -+ Hr

symk(A, M) -+ Hliek(A, M) ** H

k;yl

m(A, M)

is exact. A connected homomorphism

is induced by homomorphism

6 : ZJ^A, M) -* Zk-lm(A, M), ^ ^ drsymψ.

Proof. We will check that for k > 0 the following diagram is commutative

Cksym(A,M) ^ C

if if

For ψ G Crsymk(A,M), we have

fdrsymψ(a1,... ,ak+1) =

,a1,... ,ds,... ,ak+1) =

i , ai^,^dl,...,as,..., ak))

( a s ° ai, a 1 , . . . , a di,.., a ds,... , ak))

s o o j , a i , . . . ,ds,... ,di.... ,ak+1)

Y]

{-l)s+l+k+lip(as, a1,... ,ds,... ,di,... , dj-i, [ai, aj],... , ak+1)

(as, a1,... , c t j , , . . , as,... , ak)) o ai

s,a1,..., a ds,.., ,di— , ak+\) o ai,

and

24

Z_,\ — -U JtyyOili • • • i Oii, • • • , OLj— 1) [O-ij OLj]i • • • > ^ f c + l )

+ 53(~ 1) t[/V ;( ai)--- ,fli,--- ,ak+1),ai] =i

^ ij ^ y ^ u s , u i , . . . , a s , • • • , u%, • • • , Uj — i, [ut, Uj\,..

s, a i , • • • , a;, • • • , a V ^ a j - i , [a*, %•],-•• ,

Thus, according to right-symmetric identity,

So, a short exact sequence of cochain complexes takes place

0 -+ ®k>oCk

rsym{A,M) -* k>0Cr

symk(A,M) -* k> 0Cliek(A,M) ^ 0.rsym

In particular, a long cohomological sequenceH°rsym(A, M) -* Hrsym2(A, M) -* Hlie2(A, M) -

^ H*-2m(A, M) -+ Hrsymk(A, M) -+ Hliek(A, M)

is exact. The exactness of the beginning part

0 -+ Zrsym1(A, M) -* ZUA M) ^ H°rsym{A, M) -* Hrsym2(A, M)

we check directly. It is clear that dlieψ = 0, if drsymψ = 0, ψ G C1(A,M) . So, the natural

homomophism Z* m(A,M) —>• Zlie1(A,M) is a monomorphism. Let δφ, φ G Zlie1(A,M), gives

us a trivial class in H°(A,M) = Z^sym(A,M). Then φ e Zrsym1(A,M), since δφ = drsymφ.

Suppose that σ G Z^sym(A, M) is a coboundary in Zr

sym2(A,M), say σ = drsymω, for some

w G Crsym1(A, M). T h e n drsymω(a, b) = σ (a, b) = σ(b, a) = drsymω(b, a), for a n y o , 5 e A. T h i s

means that dlieω = 0.

The theorem is proved completely. •

3.5. Cup product in right-symmetric cohomologies.

Theorem 3.6. Assume that a cup product of A— modules U : M x N —> S is given. Then abilinear map

C%lm(A, M) x Ctie(A, N) -* C%l

m(A, S), (V, )^tpu,

defined by

Crsymk+1(A,M) x Cliel(A,N) -+ Crsymk+l+1(A,S),

ipU(f)(ao,ai,... ,ak+l) =

25

sgnσψ(a0,aσ(1),... ,aσ(k)) U φ(aσ

a G Symk+l,σ(1) < ••• < σ(k),

σ(k + 1) < ••• < σ(k + l)

is also a cup product:

{a U f3)prsym{x) = aprsym{x) U j5 + a U /3pKe(a:), Va G C;+^(A, M), V/3 e C^e(A, iV).

Moreover,

drsym(ψ U φ) = d^™^ U 0 - (-1) V U d l i eφ, (5)

for any ψ e Crsymk+1(A,M), φ e Cliel(A,N), k,l>0.

Proof. By theorem ??F(η U φ) = F(η) U φ,

F(αρlie(x)) = (Fα)ρrsym(x),

for any r? e C i ( A , C ^ ^ , M)), a G Crfc+^(A, M), e ClUe{A, N),xeA.

Prolongate the cup product M x N —>• S, (m,n) i—>• mUn, of A— modules to a cup productof Al ie -modules

C1(A,M)lie x Nl i e -»• C1(A,S)lie,

(f,n)^fUn, (/Un)(fl) = / («)Un.

Check the correctness of this definition:

((/Un)o(a))(6) =

drsym(fUn)(b,a) =

b o ((f u n)(a)) - (f U n)(b o a) + ((f U n)(b)) o a =

6 o (f(a) U n) - f(b o a) U n + (f(b) U n ) o a =

(b o f(a)) Un- f{boa)Un + (f(b) o a) U n + f(b) U [n, a] =

(drsymf(b, tt))Ufi+ f(b) U [n, a] =

Thus, we have a cup product of Chevalley-Eilenberg cochain complexes [?]

Cliek(A,C1(A,M)) xClUe{A,N) ^c£\A,C\A,S))

{{r]U(f)){ai,... ,ak+l)}(a0) =

Y sgnσ{η(aσ(1),... ,aσ(k)) U (f){a^k+1),... ,a

a G Symk+l,Σ ( 1 ) < ••• + (-l)fcr? U dZie0,

accordingly (??),drsym((Fη) U φ) = drsymF(η U φ) =

FdHe(r? U 0) = F(dHer? U 0 + (-l)fcr? U dH e^) =

26

7r](J(f) + (-l)kFri(Jdiie(f).

By theorem ?? for any ψ G Crsymk+1(A,M),k > 0, there exists η G Ck(A,C1(A,M)), such that

ip = Fη. Hence, (??) is true. •

Corollary 3.7. The cup product

C*+L{A,M) x C?ie(A,N) -»• C*fy

l

m{A,S), (ip,)^i)\j,Crsym(induces a cup product of cohomology spaces

HrJylniA M) X HL(A, N) ^ HrJym\A, S), < M [4>]) ^ [V> U ], k > 0, I > 0.

ZlsyJ^A, M) X ffL(^, iV) -> ^ m ( ^ , 5), (^, [(/>]) ^ [̂ U (/)], Z > 0.

Proof.Zr

symk+1(A, M ) U Zliel(A, N) C Zrsymk+l+1(A, S ) , k, l > 0,

+ 1 ( A , M ) U Zliel(A, N) C E ^ + i + H A S), k>0,l>0,

l,N)uBlUe(A,N)0.»

Notice that for any module M of right-symmetric algebra A and trivial A -module K there

exists a natural cup product

M x K. -• M, (m, λ) i-̂ mλ.

So, we have a pairing of cohomology spaces

HrsymiA M) x Hlie(A,]Cj —> H*sym(A,M).

In particular, H*sym(A, M) has a natural structure of module over H*ie(A,K,). As it turnedout in some cases H*sym(A, M) is a free Hfie(A, K) -module. In section ?? we will see that thisis the case, if A = glnr

sym.Denote by M antisymmetric A -module obtained from M by fa = ra — la,la = 0. One can

construct another cup product

/C x M —>• M, A U m = λm.

We use this cup product in consideration of right-symmetric cohomologies for A = Wnrsym,

section ??.

4. DEFORMATIONS OF RIGHT-SYMMETRIC ALGEBRAS.

4.1. Deformation equations. We will follow the Gerstenhaber theory of deformations of alge-bras [?]. Let A be a right-symmetry algebra over a field K of any characteristic p. Let K((t))be a fraction field for formal power series algebra K[[x]]. Extend the main field K until K((t))and construct on the vector space A ((t)) a new right-symmetric multiplication

whereHi G Crsym2(A, M), i = 0,1,2,... , a n d /j,0(a, b) = a o b.

The right-symmetric condition for /j,t in terms of //& can be regarded as the following defor-mation equations

Hi G Zrsym2(A,A), (DFR.1)

fc-i

J 3 W l i"fc-Z = -drsymHk, (DFR.k)l=1

k = 2,3,... ,

where(tp * 4>)(a, b, c) = tp(a, (f)(b, c)) - ip((f)(a, b), c) - tp(a, (f)(c, b)) + ip((f)(a, c), 6),

27

Right-symmetric deformations fit, and νt are said to be equivalent, if there exists a map

gt=g0+tg1+tg2 + --- , gk e Crsym1(A, A ) , k = 0,1,2,... ,

with an identity map g0, such that

9il{lh{9t{a),9t{b))) = νt(a,b), Va,6 G A.

In particular, for equivalent deformations /J.t,vt, should be

ν1 = Mi + drsymg1.

In other words the first deformation terms, so called local deformations will define equivalent2-right-symmetry cohomology classes [/xi] = [ν1].

Converse, suppose that there is given a 2-cocycle of right-symmetric algebra with coefficientsin the regular module, ψ G Zrsym2(A,A), with a cohomology class [ψ] G H

r

sym2(A, A). Onecan take /xi := ψ, and try to construct [ik that will satisfy deformation equations. Evidently,(DFR.1) is true. We will say that local deformation [i\ = ψ can be prolongated to a globaldeformation until k-th term, if there exist [i2,... ,/J-k, such that equations (DFR.k) are true.If this is the case for any k > 0, we will say that local deformation [i\ can be prolongated untilglobal deformation fit or, equivalently, that fit is global deformation or prolongation of [i\. Set,

fc-i

Obsk(ψ) = ̂ m* Hk-i-l=1

Notice that the definition of Obsk(ψ) depends not only from ψ but, also from the first k — 1terms of deformation.

4.2. Third cohomologies as obstruction.

Proposition 4.1. Suppose that a local deformation [i\ = ψ can be prolongated to a globaldeformation until (k — 1) -th term. Then, Obsk(ψ) G Zrsym3(A, A) and the prolongation of [i\until k -th term is possible, if and only if [Obsk(ψ)] = 0.

Proof. For α G Ck+1(A,A),β G Cl+1(A,A) define multiplications a*(3 G Ck+l+l{A,A), a ^/3eCk+l+2(A,A) by

a* β(a1,... ,ak+l+1) =k+1J2(-l){s+1)la(ai,... ,as-i,f3(as+i,... ,as+l),as+l+1,... ,ak+l+1).s=1

a — (a1,... ,ak+l+2) =

α(a1,..., ak+1) o β(ak+2, ..., ak+l+2).

Then,

ip • 4>{a, b,c) =ip* 4>{a, c,b) -ip* 4>{a, b, c), ip, G C2(A, A),

anddrsymObsk(ψ)(a0, a1, a2, a3) =

Y^ 53 sgnadass(ni l /j,s)(a0, aσ ( 1 ), aσ(2) ,aσ(3))l+s=k,l>0,s>0 a£Sym3

where dass means Hochshild coboundary operator as in associative algberas. By [[?], §7, Th.3],

dassa * j3 = a* dassf3 - dassa */3-a^/3 + /3-^a, a,/3 G C2(A, A).

Notice that

J3 m •— Us - Us •— m = Yl m^ Us- Yl m^ns = o.l+s=k,l>0,s>0 l+s-k,l>0,s>0 l+s-k,l>0,s>0

H e n c e , a c c o r d i n g t o c o n d i t i o n s ( D F R . l ) , l < k,

28

drsymObsk(ψ)(a 0,a1,a2,a3) =

l+s=k,l>0,s>0 a£Sym3

/ , / , sgnafii* 0,a,sslJ's{ci>o,Q'(T(i)>ci>a>0,s>0 a£Sym3

-sgn σ dassni * Us (a0, aσ( 1), aσ(2), aσ(3)) =

53 E sgna/j,i(dassiJ,s(ao,(

-sgn a/j.i(a0, dass/j.s (a σ ( 1 ), a σ ( 2 ) ,

-sgnadassni(ns(ao, a σ ( 1 ) ) , a σ ( 2+sgn σ dass/xi (a0, //s (aσ( 1), aσ(2)

-sgn σ dass/j,i(ao, aσ(1), /^(a0,s>0

a0, a1,a2), a3) - /J-i(drsymiJ,s(ao, a1, a 3 ), a2) + /J.i(drsymiJ,s(ao, a2, a 3 ), a1)

, drsymfis(ai,a2, a3)) + /X|(a0, drsymfis(a2, a1, a3)) - /X|(a0, drsymfis(a3, a1, a2))

-drsymfii(fis(ao, a1), a2,a3) + drsymfii(fis(ao, a 2), a1, a3) - drsymfii(fis(ao, a 3), a1, a2)

+drsymfii(ao, fis(ai, a2), a3) - drsymfii(ao,fis(a2, a1), a3)

ni{a0, Hs(ai, a 3 ), a2) + drsym,iJ,i(ao, //s(a3, a1), a2)

where

S1= E E* /xS2(a0, ai, a 2 ), a3) + //z(//Sl * /xS2(ao, ai, a 3 ), a2) - //z(//Sl * /xS2(ao, 02,03),

s 1/j,S2(ai,a2, a3)) - /^(a0,//S1 δ /xS2(a2, a1, a3))

E Ea0,a1), a2,a3) - nh-Wi * W2(a0, /xs(ai, a 2 ), a3) + //Zl 1 /xi2(a0, //s(a2, a1), a3)

We have

EE

M/isi * ^ 2 ( a o , a1, a 3 ), a2) - //z(//si * /xS2(ao, a2, a 3 ), a1)

JLS1 * // S 2(a 2,ai,a 3)) +/^(a o ,// s i * //S 2(a 3,ai,a 2))

, a2, a1), a3) - /xz(/xSl * /xS2(a0, a3, a1), a2) + //z(//Sl * /xS2(a0, a3, a 2 ),

29

-Hi(ao,nSl * 1132(0,1,0,3,0,2)) +ni(ao,nSl * // S 2(a 2,a 3,ai)) -in(a0,/iSl * Iis2(o3, a2,a1))} =

E E sgn a l * (/isi *l+s1+s2=k,l>0,s1>0,s2>0 aeSym3

and

EEl+s=k,l>0,s>0 l1+l2=l,l1>0,l2>0

* iih(iis(ao, a1),a2, a3) - \iix * l2(/is(ao, a2), a1, a3) + /ih * iih(//s(a0, a3), a1,a2)

-/ih * l2(a0,113(01,02), a3) + iih *

+ /ih * iih(a0, iis(oi, a3), a2) - /ih * in2(a0,113(03,01), a2)

-Iih * 1112(00,113(02,03),ai)+jih * in2(a0,iis(a3,a2),ai)

- jih * jih(iis(ao, a1), a3, a2) + //Zl * /xi2(iis(a0, a2), a3, a1) - /x̂ * l 2 (/x s (a 0 , a 3), a2,

+ fih * fii2(a0, a 3,ii s(ai, a2)) - fih * fih(ao^i3jis(a2,

— Iih * fiii (ao> 02,11,3(01, a3)) + inx

- Iih * l 2 (a0, a1, jis (a2,

/xi2) * iis (a0, a σ ( 1 ), aσ(2), aσ(31+l2+s=k,l1>0,l2>0,s>0

Let α, I3,-I C2(A,A). Then,

{α * (β * γ) - (α * β) * γ}(a, b, c, d) =

α(β * γ(a, b, c), d) + α(a, /? γ (b, c, d))

- a * f3(j(a, b),c,d) +a* j3(a, 7(6, c), d) - a * /?(a, 6,7(0, d))

, c ) , d) + α ( γ ( a , b), β(c, d))

a,γ(b,c)),d) -α(a, β( γ(b,c),d))

So, for any α, β, γ G C2(A, A),

a * ( / 3 * 7 + 7*/3) — ( a * / 3 ) * 7 —

For these reasons,

E E sgna/iSl * (11s

30

Y^ n σ (/ih * l2) * nh (a0, aσ

So, we prove that drsymObsk(ψ) = 0, if drsymObsl(ψ) = 0, for any 0 < l < k. •

Corollary 4.2. If Hrsym3(A, A) = 0, then any local deformation can be prolongated.

4.3. Steenrod squares. Let char k = p > 0. In this subsection we recall Gerstenhaber'sconstruction [?] of the homomorpism

Sq : Zrsym1(A, A) -* Zr

sym2(A, A), D ^ SqD

regarding the right-symmetric algebras. For any derivation D G Zrsym1(A,A) its p-th powerDp is also a derivation, Dp G Zrsym1(A,A). The proof is based on the following property ofbinomial coefficients: an integer (̂ ) can be divided into p, if 0 < a < p. Then,

Dp(a ob)- Dp(a) ob-ao Dp(b) =

53 p Di(a) ° Dp~l(b) = 0(modp).%=\ V /

In particular, we can consider integers (?)/p, 0 < i < p by modulus p and introduce 2-cocycleSqD with coefficients in the regular module

P-I

Sq D(a, b) = Y, Di(a) o Dp-\b)/i\(p - i)!.

This cocycle is called the Steenrod Square of derivation D and can be interpreted as an ob-struction to prolongation of derivation to automorphism.

5. CALCULATIONS

5.1. Standard 2-cocycles of right-symmetric algebras. In this subsection we will give thesecond interpretation of the identity drsym3m = 0, m G M, mentioned in section ??.

Proposition 5.1. i) Let M be a module over right-symmetric algebra A and M is its sub-module. Suppose that for fh G M,

(fh,a,b) G M, Va, b G A.

Then 2-cochain ipm G C2(A,M) defined by

ipm(a, b) = fh o (a o b) — (fh o a) o b,

is symmetric 2-cocycle, ipm G Z2(A,M).

If fh o a G M, Va G A, then [ipm] = [m(0;b) = ao (fhob).

Notice that in the denotions of section ??, ipm = V(m).Proof.

ipm(a, b) = (m, a, b) = (m, b, a) = ipm(b, a),

arsymYrhyd] v? Cj —

ao (fh, b,c) — ao (fh, c, b) — (fh, ao b, c) + (fh, aoc, b) + (fh, a, [b, c]) — (fh, a, b) o c + (fh, a, c) o b =

— (fh, a o b, c) + (fh, a, b o c) — (fh, a, b) o c

+(fh, a o c, b) — (fh, a, c o b) + (fh, a, c) o b =

fh o (a, b, c) — (fh o a, b, c)

—fh o (a, c, b) + (fh o a, c, b) =

0.

31

If m o a G M, then we can introduce a linear map ω : A —> M, a i—> m o a. We obtain

ipm(a, b) + drsymω(a, b) =

ipm(a; b) — ω(a o b) + ω(a) o b + a o ω(b) =

ipm(a; b) — fh o (a o b) + (m o a) o b + a o (m o b) =

ao (mob).

In other words, ao6, and (a, b) i-> a * b, such that the following conditions hold

a o (b o c) — (a o b) o c — a o (c o b) + (a o c) o b = 0,

a * (b * c) — b * (a * c) = 0 ,

a o (b * c) — b * (a o c) = 0 ,

( a * 6 - 6 * a - a o 6 + 6 o a ) * c = 0,

(a o b — b o a) * c + a * (c o b) — (a * c) o b — b * (c o a) + (b * c) o a = 0.

In particular, A is a right-symmetric algebra.Let Zl(A) be the left center of A, Ql(A) is the left units space and Nl(A) = Zi(A)®Qi(A)

is the left semi-center.

Theorem 5.2. For A = Wnrsym, if p = 0, or A = Wn(m), if p > 2,

~1 : 0 < ki < mi,i = 1,... ,n} =

r s y m 1 ( A , A ) = { a d d i } a d x i d j , δ ( p > 0) 0) = 0, if p = 0 and =1, if p > 0.

Proof. Any derivation of right-symmetric algebra A induces a derivation of Lie algebra

drsym(a, b) = 0, Va, b e A => dlief(a, b) = drsymf(a, b) - drsym(b, a) = 0, Va, b e A.

The corresponding homomorphism Zrsym1(A,A) —> Zlie1(A,A) is monomorphism. It is known,

that all Lie derivations of Wn are inner, i.e. have a form adudi, where u &U,i = 1,... ,n. In

case of Wn(m),p > 0, outer derivations 9f *, 0 < ki < mi, i = 1,... , mi, appear. Since

ada(bo c) — ad a(b) o c — b o ad a(c) =

a o (b o c) — (b o c) o a — (a o b) o c + (b o a) o c — b o (a o c) + b o (c o a) =

a o (b o c) — (a o b) o c,

Lie derivation ad a : b i—> [a, b] := a o b — b o a, is a right-symmetric derivation, if and only if

a G Al.a s s.

It is easy to see that,

o (vdj o w9 s) — («9j o vdj) o w9 s = djds(u)vwdi.

Therefore,

In case of p > 0, direct calculations show that c% z G Z^sym(A,A). Other statements of thetheorem are evident. •

32

5.3. Pairing of W^sym— modules and cocycle constructions.

Theorem 5.3. Let A = Wn,p = 0, or A = Wn(m),p > 3. The space Hrsym2(A,A),p = 0, has

a basis consisting of cocycle classes of four types ψs,l,r1,ψl,r2,ψs,l3,ψl4, s, l, r = 1,... ,n, such that

vdj) = (5i>sudj(v)di -xsdi(u)dj(v)di),

tpf(udi,vdj) = di(u)dj(v)di.

rsym2(In the case of p > 3, the space Hrsym2(A,A) has a basis with cohomological classes of thefollowing cocycles of five types.

l v d j " > = 5j,rx'flr~1(5i>suvdi - xsdi(u)vdi),

di,vdj) = (5iySudj(v)di - xsdi(u)dj(v)di),

where s, l, r = 1,... , n, 0 < kl < ml.

The proof is based on the following observations.

Suppose that A— module M preserve the action of Nl(A) :

zom = 0,Vz G Zl(A), e o m = m,Ve e Ql(A),

for any m G M. Define an operator

Crsymk+1(A, M) —> Crsymk(A, M ) ,

i0(a)ψ(a1,... ,ak)= ψ(a0, a1,... ,ak)

and an operator

1 C rs

k

Tψ(a0, a1,... , ak) = ^2(-iy+ka,i * ψ(a0, a1,... , a i ; . . . , a k ) .

i=1

Then

1 drsym = drsymT ,

and for any a e Nl(A),ψ e Zrsymk+1(A,A),

i0(a)ψ G Zliek(A,Aanti).

Define a pairing of regular A— module A and antisymmetric A— module U

AxU^A, udiUv = uvdi.

Notice that,

Aanti^USZiiA).

In particular, we have pairing

A x Aanti —> A, udi U vdj = uvdi.

Therefore we have imbedding

Zrsym1(A, A) x Hliek(A, U) -* Hrsymk+1(A, A).

33

Notice that the four types of cocycles mentioned above can be obtained from Zrsym1(A, A)

(see section ??) and Hlie1(A,U), by pairing ψ U φ, ψ e Zrsym1(A,A), φ e Hlie1(A,U). Recall

that Hlie1(A, U) can be generated by the classes of cocycles udi i-> ux~l5i>r, and udi i-> di{u).

Another interpretation of cocycles of types 1 and 2 can be given in terms of standard cocycles(see section ??). For simplicity consider only the case of p = 0. We have

ipl>l>r = dujSti>r,for uoSti>r{udi) = Inxr[xsdi,udi],

ip\r = dωl,r,for u)itr{udi) = \nxrdi(u)di,

and

V(xslnxrdi)(udi,vdj) = didj(xs\n xr)uv,

V(ln xrdi)(udi,vdj) = -5itr5jtrv,vdi.

Therefore,

s,l,r1] = Mxalnxrdi)],

because of

%) _ drsymuls>l>r, for u

l

s>l>r e Cl

rsym{Wn, Wn), ul

s>l>r{udi) = Si^xsudi,

•di) - drsymujfr,ioi ijj\r e C}sym(Wn, Wn), ujf>r(udi) = 5i}rx~ludi.

Theorem 5.4. Let A = W1rsym, if p = 0, and A = W1(m), if p > 3. Then Hrsym2(A,A) hasdimension 4, if p = 0, and cohomological classes of the following cocycles generate a basic

ipl(ud,vd) = x~luvd,

ip2{ud,vd) = x~ld{u)v d,

ip3(ud,vd) = (u- xd{u))d{v)d,

ip4(ud,vd) =d(u)d(v)d.

(Recall that, d = dx, for n = 1.).For A = W1(m),p > 3, the group Hrsym2(A, A) is (3m+ 2)— dimensional and the classes of

the following cocycles generate its basis

ipl(ud,vd) = xpm-lcfk(u)vd, 0

34

5.4. Right-symmetric central extensions of Novikov algebras. Let A be a Novikovalgebra, R e Der0 A := {D e Der A : ao R(b) = bo R(a), Va, b e A}, and π : A —K , a linearmap, such that

π(R(a)) = 0 , Va e A.

Define ψ e Crsym2(A,K), by

#(a,6) =ir(aoR(b)).

Lemma 5.5. $ G Z?sym(A,lC).

Proof. Since, ao R(b) = bo R(a), then

35

So, ^ ^ J A , / C ) . .In particular, algebras

A = W1rsym = {ei : ei o ej = (i + 1)ei+j, i, j eZ},p = 0,

A = W1rsym(m) = {ei : ei o e j = I I e i + j , —1 0.

Theorem 5.6. Let A = W1 r s y m, if p = 0 and A = W1(m), if p > 0. Then the secondright-symmetric cohomology space Hrsym2(A,K) has dimension 1 and generates by a class ofcocycles

#(ej,ej) = —(j + I)j5i+jt-i, if p = 0,

fy (P • P • \ ( 1 V A ' , • rn 1 7f n ^ P l( y V O ^ , O i I V -L y t \ l IP — i- 1 J I ^ '

If A is considered as a Novikov algebra, then any Novikov central extension is split: Hnov2(W1, K) =0.

Recall that Novikov cohomologies are defined in [?]. Central extensions of Cartan Type Liealgebras are described in [?]

Proof. For u G U = K[[x±1]] let π(u) be its coefficient at x~l. Then vr( , (u,v)»ir(u-v),

is compatible with the action of Alie :

TT(—(U O a) • v — u • (v o a) — u • (a o v)) =

vr(-9(a • (u • v))) = 0,

for all a e A , u e [ / o , D 6 t / i . So, we have a pairing of Alie-modules ( , ) : U0 X U 1>• . Thispairing is nondegenerate. Thus, a dual Al ie-module to U0 is U1. Since,

(f o a)(b) = drsymf(b,a) = -f(bo a), f e C1(N,k),a,b e N,

we see that the Alie -module C1(A,K) is isomorphic to the dual of U0. This ends the proving

By theorem ??

Hrsym2(A, K) = Hlie1(A, C1(A, K)) = Hlie1(W1, U1).

36

By the resutls of Gelfand and Fuchs [?] the space Hlie1(W1, U1) is 1-dimensional and generatesby a class of cocycle a i—> 2(a). An analogous statement is also true in the case of p > 0 [?].The corresponding right- symmetric cocycle is the cocycle #.

If ψ : Ax A —>• K is a cocycle for central extension in the category of Novikov algebras, then

ψ(a, b o c) - ψ(b, a o c) = 0, Va,6, C A .

The algebra A = W1n°v has an element e0 that has the property e0 o c = c, for any c e iTake a : = e0. We have

ψ(e0, boc) = ψ(b, e0 o c) = ψ(b, c), V6, c e A .

Therefore, for ω G Cnov1(A, K) = C1(A,K), such that ω(a) = —ψ(e0,a), we have

ψ(b, c) = -w(6 o c) = dnovω(b, c).

Recall that, dnovφ = drightφ, for any φ e C1(A, K). So, any 2-cocycle of W1nov (in sense of

Novikov) with coefficients in the trivial module, is a coboundary. •

5.5. Cohomologies of Wnrsym in an antisymmetric module. Recall that a multiplicationi n Wnr

sym is g i v e n b y ar\ o bdj = bdj(a)di, w h e r e a,beU = K[[x±1,... ,x±1]]. E n d o w U b ya structure of antisymmetric Wnr

sym -module: u o adi = adi{u).Let Vtn = {udx 1 • • • t\dxn : u G U} be an antisymmetric Wnr

sym -module of n -dimensionaldifferential forms:

{u dx\ A • • • da; ra) o adi = di{au) dx\ A • • • A dxn.

Let M be V ^ ^ m -module. Construct a cup product of Wnrsym -modules

[ / x f l n ® M ^ M , ( u U t ) & i A ••• Adxn m) =7r(uv)m, (8)

where π(u) for u G U denotes a coefficient of u at x~[l.. .x~l. Recall that M is an antisym-metric A -module corresponding to M, such that fa = ra — la, la = 0. Notice that

π(deri(u)) = 0 , Mu G U, i = 1,... , n.

Therefore,

(u o a9j) U (f dx\ A • • • da;ra ® m ) | « U [ ! i da;i A • • • dxn m, adi] =

adi{u) U (v dx\ A • • • A dxn ® m ) | « U (di(av) dx\ A • • • A cfccra ® m)

+ u U v dx 1 • • • A dxn ® [m, adi] =

n(adi(u)v + udi(av)) m + π(uv) [m, a9j] =

n(di(auv)) + π(uv) [m, a9j] =

( U U D dx\ A • • • A da;ra) ® [m, adi] =

( t i U o da;i A • • • A dxn m) o a9j.

and the definition of the cup product (??) is correct.

Theorem 5.7. Let M be an antisymmetric W^"1 -module. The cup product ofmodules (??) induces an isomorphism

(Wn, M) = Zrsym1(Wn, U) k(Wn, Q™ ® M), k > 0.

An isomorphism takes place

Zrsym1(Wn, U) ^ Brsym1(Wn, U) = {dxi : i = 1,... , n} ^ A1.

37

Proof. We will argue as in the previos subsection. For a Wn -module M its dual moduleis denoted by M'. Consider A1 = {dx1,... ,dxn} as a trivial module over Lie algebra Wn.Endow U 0.•

Corollary 5.8. ^ rf c + ^ ( ^ r a , K) = A

1 eg) ^ e ( W r a , Qra), A; > 0.

Recall that Hfie(Wn,tt*) is calculated by Gelfand and Fuchs [?].

5.6. Cohomologies of glnrsym in an antisymmetric module.

Theorem 5.9. Let A = glnrsym, chark = 0. Let M be a finite-dimensional A-module, suchthat a Alie -module M is a tensor module. Then the cup product M x /C —> M, m U λ = λm,induces an isomorphism

Hkr+l

m{A, M) - Z r ^ m ( A M) ^ e ( A , /C), A; > 0.

Proof. By theorem ??,

Hkr£m{A,M)^Ht{A,C\A,M)), k > 0.

By theorem 2.1.2 of [?],

It remains to notice that,

C1(A, M)Alie := {f G C1(A, M) : [f, a] = 0, Va G A}

is exactly Zrsym1(A,M). It is evident:

[f,a](b) = (foa)(b)=drsymf(b,a), V o , 6 e A •

Corollary 5.10. Let A = glnrsym and M be a irreducible antisymmetric A-module. ThenHrsymk(A, M) / 0, k > 0, if and only if M = A, and

Hrsymk+1(gln, (gln)anti) = H^igln, K), k > 0.

In particular, Hrsymk(gln, K) = 0, k > 0.

Proof. Since M is antisymmetric,

Zrsym1(A, M) = {f:A^M:f(boa) = f(b) o a } .

Thus, any f G Zrsym1(A, M) gives us a homomorphism of right modules f : A —>• M. Since

right modules M and A are irreducible, by the Lemma of Shur, Zrsym1(A, M) = /C, if M = A,

and Zrsym1(A,M) = 0, if M ^ A. •

38

5.7. Cohomologies of glnrsym in a regular module.

Theorem 5.11. Let A = glnrsym over a field K of characteristic 0 and M = A be its regularmodule. Then the cup product M x /C —>• M, m U λ = λm, induces an isomorphism

Hrsymk+1(A, M) ^ Zrsym1(A, A) Hliek(gln, K), Zrsym1(A, A) ^ sln, k>0.

In particular, any cocycle class in Hrsymk+1(gln,gln) has a representative that can be presented as

adXUtp, where ψ e Zliek(gln, K).

Proof. Any right-symmetric 1-cocycle of an associative algebra A is also an associative 1-cocycle and converse, any associative 1-cocycle is a right-symmetric 1-cocycle. So, Zrsym1(gln,gln) =Zass1(gln,gln). Any derivation of the associative algebra gln is a derivation of the Lie algebraof gln. Any derivation of gln

li e, except ai—*tra, is inner. So, the following sequence is exact

0 -> Zrsym1(gln,gln) -> Zlie1(gln,gln) - . K -^ 0.

In particular,

Zrsym1(gln,gln) = {adX : X G sln} ^ Zlie1(sln, sln) ^ sln.

It remains to use theorem ??. •

Corollary 5.12. An algebra gln as a right-symmetric algebra has nontrivial deformations. Anyright-symmetric local deformation (2-cocycle of the regular module) is equivalent to a 2-cocycler]x of the form

ηX(a,b) = (trb)[X,a],

for some X e sln. Any local right-symmetric deformation can be prolongated. Any formalright-symmetric deformation of gln is equivalent to a deformation of the form

Ht(a,b) =aob + t(trb)[X,a], X e sln,

where (a, b) i—> a o b is a usual associative multiplication of matrices.

Proof. These statements can be obtained from the following cohomological facts

ln,gln) = Zright1(gln,gln) (8) lie1(gln, K) = {adX Utr:Xe sln},

Hrsym3(gln, gln) = 0 ,

and corollary ??. •

Remark. For Ω G C1(gln,gln), ω(a) = (tra)X, we have

(η + drsymω)(a,b) = (trb)[X,a] + (trb)aoX - (tra o b)X + (tra)X o b = fjx(a,b),

where

Vx(a, b) = (tr b)X oa-(trao b)X + (tr a)X o b.

Therefore, [ηX] ~ [vx]- Notice that fjx is a symmetric cocycle:f](a,b) =fj(b,a).

The prolongation formula for fjx is a little bit complicated. It can be given by

fH(a,b) = tl(M$t(a),t(b))),

where

Φt = id + tω.

In particular,i\a) = a-t(tr a)X + t 2 (tr a)2X2 ,

and some of the beginning terms of jit look like

jk{a,b) =aob + tX o((tra)b- (traob) + (trb)a)

+t2 (tr atrb-(trao b)2X2 - (tr atr(Xo b))X - (tr (a o X) tr b)X) + • • •

39

So, right-symmetric prolongation can be constructed in a such way, that corresponding Liemultiplication will not be changed:

p,t(a,b) -p,t(b,a) = [a,b].

Acknowledgements. The author is deeply grateful to INTAS Foundation (Prof. U. Rehmann),Bielelfeld University (Prof. K. Ringel) and the Abdus Salam International Centre for TheoreticalPhysics, Trieste, Italy (Prof. M. Virasoro and Prof. M.S. Narasimhan) for support.

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