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Poincaré series of quantum matrixbialgebras determined by a pair ofquantum spacesPhung Ho Hai aa Section Physik der Universitat Munchen Theresienstr 37 , 80333,Munchen E-mail:Published online: 27 Jun 2007.

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COMMUNICATIONS IN ALGEBRA, 23(3), 879-890 (1995)

POINCARE SERIES OF QUANTUM MATRIX BIALGEBRAS DETERMINED BY A PAIR OF QUANTUM SPACES

Phung Ho Hai Section Physik der Universitat Miinchen

Theresienstr 37, 80333 Miinchen e-mail: phung@lswes8.ls-wess.physik.uni-muenchen.de

Abstract

The dimension of the third homogeneous component of a matrix quantum bialgebra, determined by pair of quantum spaces, is calculated. The Poincar.4 series of some deformations of GL(n) is calculated. A new deformation of GL(3) with the correct dimension is given.

1 Introduction

Quantum groups are a subcategory of the category of Hopf-algebras, which are generally non-commutative and non-cocommutative and possess many properties similar to those of classical universal enveloping algebras or, by duality, function rings of affine algebraic groups. There are many approaches to quantum groups, the "R-matrix" and Quantum Yang-Baxter equations approach, which was introduced by Faddeev, Takhtajan, Reshetikhin and others, formal deformations of commuta- tive or cocommutative Hopf algebras especially of the universal enveloping algebra U ( g ) of a Lie algebra g, which was systematically studied by Drinfeld, Jimbo and others. Quantum groups can be also considered as the symmetries of quadratic non- commutative or quantum spaces. This construction was given by Yu.Manin [M2]. The methods in this paper are based on Manin's construction.

A family of quantum spaces defines an algebra, which is universally coacting upon the spaces of this family. This algebra because of its universality is a bialgebra. We call it a quantum matrix bialgebra (QMB). To obtain a Hopf algebra envelope of this bialgebra, one often has to invert a certain quantum determinant, which is a group like element and in many examples a central element In this paper we restrict ourselves to an investigation of the quantum matrix bialgebra, which induces deformations of GL(n) . More concretely we study a quantum matrix bialgebra, which is determined by two quantum spaces, which generalize a pair of symmetric and antisymmetric tensor algebras on a linear space. It is natural to consider the

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Copyright O 1995 by Marcel Dekker, Inc.

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above QMB as a deformation of the bialgebra M ( n ) (a polynomial ring on the semigroup of linear transformations of a vectorspace). Since our bialgebra is a graded algebra, the question about its Poincark series arises.

Note that in our construction we obtain an operator - "R-matrix". We always require it to satisfy the so called quantum Yang-Baxter equation. This operator plays an important rble in quantum theory. For deformations of M(n) it should satisfy the Hecke equation. In the undeformed case, for M(n), it is just the twist operator.

Recently several authors have considered the polynomiality and the Poincark series of a quantum matrix bialgebra by using the diamond lemma. The troubles usually happen on cubic monomials. In general, the method of the diamond lemma seems not to be effective because there is no sense of reduction system. In this work we do the first step of calculating the Poincark series without using the diamond lemma. We use the method developed by Sudbery[SP]. We get a formula for calcu- lating the dimension of the third component of the QMB via the dimension of the components of the quantum spaces (Section 3). Nevertheless, we remark that the obtained fornlula is independent of the matrix R. It seems to be complicated to do the same for the higher components of the QMB. It should be interesting to show that the same independence takes place for all higher homogeneous components. If this would be true, we could say that if the quantum spaces have the correct Poincark series then the QMB has the correct Poincark series, too.

On the other side, knowing the dimension of the second and third components in some cases we can already obtain the Poincark series of the QMB. Applying Bondal's theorem ([Bo], Theorem 3) to M ( n ) we have, if R = R(q),q E R is some continuously parameterized deformation of P - the twist operator, R(1) = P, and if the Poincark series of the quantum spaces are correct, then for every interger n we can choose a neighborhood of 1, such that the dimensions of the homogeneous components of the QhlB obtained from R(q) with q in this neighborhood is correct up t o the n-th component.

In section 4 we investigate the polynomiality of all known QMBs which give a deformation of GL(n). The polynomiality of the multiparameter deformations was shown in [ATS]. We show here the polynomiality of a deformation obtained by Creme and Gervais [CG]. At the end of the section we give a new deformation of GL(3).

2 A Yang-Baxter operator on a universal bialgebra.

Let k be a fixed algebraically closed field, V be a k-vector space of finite dimension n. A quadratic algebra A is a factor algebra of the tensor algebra T ( V ) by a two-sided ideal generated by a subspace RA of V @ V. Hence A is a graded algebra:

We call the pair (A, V) a quantum space. Let B be a k-algebra. A left coaction of B on the quantum space ( A , V) is an

algebra homomorphism b B : A - B @ A

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A coaction of B on a family of quantum spaces (A,, V), a E J is a family of coactions of B on each (A,, V), a E J, such that their restrictions on V coincide.

An algebra E is said to universally coact upon (A,, V) if for every algebra B coacting on (A,,V),a E J, there exists a unique algebra homomorphism q5 : E + B such that (4 @ id)& = 6k,a 6 J, where 6,, 6: denote the coactions of E and B respectively. Mukhin [Mu] and Sudbery [Sl] show that E exists. The bialgebra structure on E follows from the universality.

Let 6 denote the isomorphism

which interchanges the 2nd and 3rd components. Analogously we have the isomor- phism of the two spaces (V*)@3 @ VB3 and End(V)@3, which is denoted by the same 0.

In this paper we study certain deformations of M(n). In this case there are only two quantum spaces ( A , V) and (S, V) with RA $ Rs = V@*. ( A , V) and (S, V) play the analogous roles to the antisymmetric and symmetric tensor algebras on the space V. Let E be the universal matrix bialgebra coacting on (A, V) and (S, V). Then E is a quadratic algebra over El = V* @ V. The relations on E are given by RE = 6(Ri @ Rs + Rfi @ RA).

Let PA and PS be the projections onto RA and Rs respectively with respect to RA $ Rs = Vm2. Let us consider for arbitrary q E k, q + -1, the operator

and R* : V* @ V* -+ V* @ V*. R and R* are plainly diagonalizable. Let t;, i = 1,. . . , n2 be the eigenvectors of R which form a basis on V @ V, the dual basis to it consists of eigenvectors of R*, denote them by u;, i = 1, . . . , n2. The vectors u; @ t j are eigenvectors of R* @ R-' and form a basis on V* @ V* @ V @ V. Consequently

We assume that our construction is Yang-Baxter, in other words, there exists a q such that R satisfies

R is called a Yang-Baxter operator. And we assume that q obeys the following condition

B(P + 1)(q2 + P + 1) # 0. (3) For a quadratic algebra A we have A = $goAi. Since the A; are all finite dimen- sional over k we can form a formal series PA(t) = C g O ( d i m k ~ ; ) t i , which is called the Poincark series of A.

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Definition. a ) A pair of quadratic algebras ( S , V ) and ( A , V ) is said to be poincarb if their Poincark series coincide with the ones of the symmetric and antisymmetric tensor algebras over V respectively. b) The bialgebra E is said to be poincark if its Poincare series coincides with the one of the polynomial ring on n2 parameters.

3 Yang-Baxter operator with Hecke equation, The main theorem

The aim of this section is to calculate the dimension of the third homogeneous component of E via the dimensions of components of A and S. Thus, let si = dim S,, Xi = dim Ai, i = 2,3 be given. We want to find e3 = dim E3.

Let R = qPs - PA be Yang-Baxter and let q satisfy (3) . Then ( R - q)(R+ 1) = 0. Consider the following operators

P, = P,(R) = (R12 - a)(R23R12 - aR23 + a2)(a = -1, q)

Pr = Pr(R) = (R12 - q)(R23 + 1)

PL = K ( R ) = (R23 - q)(R12 + 1)

Then R13PL = PrRI3, where R13 = R12R23R12 = R23R12R23. By I Ia we denote the subspaces Imp, of VB3, ( a = -1, q , ~ , r ) . The notations L and r are due to Sudbery, they are chosen for their resemblance to the Young tableau.

Analogously we define P$ = P,(R*) and II$ = Imp:, a = 1, g ,~ , r . R* is the adjoint operator of R.

For a = -1,q we have

The following lemma is due to Sudbery.

Lemma 3.1 ([SZ], Lemma 2) There ezist constants c, such that c,P, are pmjec- tions onto II, and $11, = V B 3 , a = -1, q , r , ~ .

From the definition we have

We claim KerP, = Im (R l z + 1) + Im (R23 + 1).

Indeed if x E Ker P, : (R12 - q)(R23R12 - qR23 + q2)z = 0, then

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Hence, if (1 + q)(q2 + q + 1 ) f 0, x E Im (Rlz + 1) + Im (RZ3 t 1). Conversely, if x E Im (Rl2 + I ) + Im (R23 + I), then x = xt $ z", where xt E Im ( R n f l ) , xJJ 6 Im (R23+1). And obviously P,x = Pqzt+ PqxU = 0. Consequently

s3 = dim Im P,, X 3 = dim Im PI. ( 6 )

We use the result obtained by Sudbery. For x E I& put y = q-lR13x then y E & and

R12x = -2 RZ3x = Q X - y Rlzy = -qx t qy R23~ = - Y (7)

The first and last equations are obvious. According to (4), (Rl2RZ3-qRZ3 tq2)Pr = 0, this shows the third equation R23x = qx - y , the second one follows from it. Analogously we have, for x+ E II;-+ and y+ = q-lRi3x, then y+ E IT? and

Now we consider the operator = B(R* @ ~ - l ) B - l . It this easy to check that

By the isomorphism 0 : ( v * ) @ ~ @ V @ ~ + (v*@v)B3 we can regard xlz as R ; ~ @ R;; acting upon (v*)g3 @ VB3. Consequently a is a Yang-Baxter operator.

By considering the eigenvectors of R which form a basis on V @ V we get (2 - 1)(B + q)(B + p-') = 0 and dim Im (3 - 1) = 2X2sz. Since X2 f sz = n2, by using ( 1 ) we have

e2 = dim E2 = n4 - dim Im (x - 1) = X i + s;. (9) - - - - - - -

Lemma 3.2 Let 9 = R12 R23R12 - R12R23 - R23R12 + X I 2 + x23 - 1, then

Proof. Since 2 is a Yang-Baxter operator

Hence Im @ C (Im ( 8 1 2 - 1) Im (&3 - 1)) = W. If we can show that W n Ker 9 = 0, then dim W f dim Ker @ 5 n6(since

they are subspaces of (V*)g3 @ I f B 3 ) , But dim Im @ $ dim Ker @ = n" therefore dim W = dim Im @, which means W = Im @. - - - - - -

Assume L := W n Ker9 # (0). Denote X I ~ = R12R23R12 = R23R~2R23, then

- - - - - - - - since R13R12 = R23R13, R12R13 = R13& Hence L is invariant under zl3, therefore there exists a vector x # 0 in L so that 3 1 3 2 = XZ.

If X = -1 then

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Since x E W = 1m(X1z - 1) n Im(& - I), we get (x12 + q-1)(X12 t q)x = 0 and (3;; + xl2 + q + q-l)x = 0. The same equation holds for Z23. Thus

Hence @ z = - 2 (q + q-' + 1)x = 0, q + q-' + 1 # 0, therefore x = 0, which is a contradiction. - - -

Let X # -1, Since Z13~ = R23R12R23x = Ax we get

According to ( l l ) , (Xzi3' - 8z3)x = -X(q + q-')x - (A + 1)&3~. And since x E W c ~ m ( ~ ~ ~ - 1) we get 3 2 3 2 E 1m(al2 - 1). Therefore we get

- - Then (812 + q)(Xl2 t q-')z = 0 and ( x l z - 1)r = (Z12 - 1)(R23R12 - Z23 + 1)x = 92 = 0 hence z = 0, that means

Analogously we have - - (RI2R23 - f 1 ) ~ = 0.

- - - (14)

Multiplying (13) by xlz from the left and adding (14) we get Xl3x = R I ~ R ~ ~ R ~ ~ ~ = -x which is a contradiction. Thus we must have L = {O), which means Im@ = W. Q.E.D.

Corollary 3.3 Im @ $ Ker 9 = (V*)@3 @ vB3. (15)

Indeed Im @ n Ker @ = (0).

Proposition 3.4 Let II = IIr $ nL , II+ = llf $ II? then i)IIZl tg II,, I I t @ ILl are subspaces of Im @. ii)n; cg II,, II: @ II, II+ @ II,, ( a = -1, q) are subspaces of Ker @. i i i ) The subspace II+ @ I I is invariant under the action of @ and decomposes into

a direct sum of 4-dimensional @-invariant subspaces. In each of them the operator @ has rank 1.

-(q-I + 1)(q-2 + q-' f 1)x @ y, therefore z @ y E Im a. i i) If x @ y E II: @ II,, ( a = - 1 , ~ ) then x12x tg y = & 3 ~ 8 y = x @ y SO that

@ x @ y = O . For x @ y E II?, @ II,R;,x = Ra3x = qx and P I ( ~ R - ' ) ~ = = 0 hence

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P O I N C A ~ SERIES OF QUANTUM MATRIX BIALGEBRAS 885

iii) According to (7) and (8) if xj form a basis in TI,, xf form a basis in IIT then y j = q-'R13xj form a basis in II,, y: = q-1R13zt form a basis in II:, and the subspaces of IIt @3 II spanned by the four vectors x t @ x3, x+ @3 yj, y: @ xj , y: @ yj, are @invariant. This shows the first part of iii). The second one can be proved by direct computation. Q.E.D.

Theorem 3.5 If the operator R satisfies the Yang-Baxter equation (Z), with q sa- tisfying condition (3), then we have

Proof. From (11, the relations on E3 are Im ( x l z - 1) + Im (z23 - I) , hence dim E3 = n6 - dim (Im (Biz - 1) + Im (& - 1)) = n6 - dim Im (Xl2 - 1) - dim Im (zz3 - 1) + dim (Im (xlz - 1) n Im (323 - I)) = n6 - 4n2szX2 + dim Im @ (according to (5) and Lemma (3.2)). Since II $ II-1 $ IIp = VB3, we have

According to Corollary (3.3) and Proposition (3.4)

dim Im cI, = dim (II?, @ Il,) + dim (IIT @ II-,) + dim (TI+ @3 n ) / 4 (18)

Indeed, (17) is a decompositon of (V*)B3 @ VB3, according to iii) in Proposition

(3.4), rI+ @ IT = (II+ 8 II n ImQ) $ (II+ 8 II n KercI,)

is a decomposition of TI+ @ IT. And dim(II+ @ II n Im@) = dim(II+ @ n) /4 , which implies (18). From Lemma (3.1) and (6) one follows

dim IT-, = ,A3, dim II, = s3, dimII = n3 - s3 - X3.

Thus dim Im Q = 2A3s3 + (n3 - sg - X3)2/4 which gives (16). Q.E.D. Theorem 3.5 shows that if the construction is Yang-Baxter then the dimensions

of Ez and E3 do not depend on the choice of the matrix R and depend only on A, and s,. If the same independence would take place for all the higher componmts, we could directly calculate the Poincare series of E in many cases, avoiding the use of the diamond lemma, which is not always applicable.

In the next section we investigate the polynomiality of some deformations of GL(n) however by using the diamond lemma. The following Corollary is useful.

Let El be spanned by {z : ) , define an ordering of 2: by Z: < zf, if either i < b or z = k,j < I . Call a monomial z in t: normally ordered if for any z ' , z" E (2;') in this monomial, z'< z" iff 2' is to the left of 2". We considcr the lexicographic order in the polynomials in z:, which corresponds to this order.

Corollary 3.6 If the dimensions of the second and third homogeneous components of S and A coincide with the ones of the symmetric and antisymmetric tensor al- gebras over V respectively, then the dimension of the 2nd and 3rd homogeneous components of E coincide with the ones of the polynomial ring on n2 parameters.

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Moreover, if every monomial, which is not normally ordered, can be represented as a sum of normally ordered monomials of strictly smaller order (in the lexicographic order), then the normally ordered monomials form a basis of E .

Proof. If s2 = n(n + 1)/2, X 2 = n(n - 1)/2

s3 = n(n + l ) (n + 2)/6, Xg = n(n - l ) ( n - 2)/6

then, by using (9),(16), we have

If the last condition ot the Corollary is verified then z:rk,z: < zk generate E2. According to (19) they must be independent and so form the basis on 152. The reordering procedure of monomials of higher power is possible but not unique. Manin has shown [All] that the different possible procedures will lead to the same result if they do so for cubic monomials, in other words if the cubic normally ordered monomials are independent. But they are so, according to (19). So that we can apply the diamond lemma for the algebra I?. Q.E.D.

Remark. In general the normally ordered monomials will not span E, even if the subspace Ez is spanned by normally ordered monomials. If they span E2 they may even not span ES. An example is the following quadratic algebra, A = k < x , y > /yx = x2 + y2 $ xy. The normally ordered monomials span A2 but do not span A3. Rote that, however its Poincare series are correct [EOW], p. 1. The next section co~~siders this question for the deformations of G L(n).

4 Polynomiality of some deformations

There are mainly three series of deformation of GL(n). Artin, Schelters and Tate [ATS] found the multiparameter deformation, which includes Sudbery's and Manin's deformations [S], [DMMZ]. Cremer and Gervais[CG] found a deformation, which is associated with the non-linearly extended Virasoro algebras. The Jordan deforma- tion of GL(2) was found by Manin[DMMZ] and was systematically studied by Wess and others [EOW]. In this section we investigate the polynomiality of the defor- mation obtained by Creme and Gervais. At the end of the section we form a new deformation for GL(3).

4.1 The deformation of E.Cremer and J.-L.Gervais.

This deformation is given by the matrix

Note that R = P . p, where p is the operator defined in [CG], p.625, P is the usual twist operator. R satisfies the Yang-Baxter equation and the equation

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P O I N C A ~ SERIES OF QUANTUM MATRIX BIALGEBRAS 887

We assume that q # fi. The subspace relations Rs and RA can be spanned by the vectors

where if i + j odd

respectively. We use the following lemma for investigating the PoincarQ series of S and A.

Lemma 4.1 ([G]p.556) Let Ps(t), PA(t) be the Poincare' series of S and A respec- tively. If the matrix R satisfies the Yang-Baxter equation (21, then

Theorem 4.2 The quantum matrix bialgebm E determined by S and A, described in (21) and (22), is pincare'

Proof. Denote the Poincark series of A and S by PA(t) and Ps(t) respectively. According to lemma (4.1) we have PA(t)Ps(-t) = 1. From (22) we have PA = (l+t)". Hence Ps(t) = &.

As we have seen in section 2, the subspace of relations on E is Rg 8 Rs $ R: @ RA when El @ El and V* Qi V* Qi V @ V are identified.

The subspace Rs is spanned by

We show that Rg can be spanned by vectors of the type

In fact, we must find coefficents c:, ( j 5 i), so that (ui j jukl) = b i b : , ( k < 1) . Fix i and j then for every composition of c;j, (uiJlukl) = 0 if either k + I # i + j

or k > j. Hence we have to find cp obeying the following system of equations

whose rank is min(j-1,n-i). Thus we get linear equations for ckJ, where the number of parameters equals the number of equation, and what is more, the coefficents a: of c:J in the s-tli equation is equal 1 if r = s , and 0 if r < s. Hence (26) and (25) have a unique solution.

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Using (21-25), we get the following relations on E:

By induction (27), (28) can be written in the following form

And so (29 ) , (30) are generating relations for E. According to the diamond lemma normally ordered monomials span E. Finally, using corollary (3 .6 ) , one finds that E is poincar6 Q.E.D.

4.2 The Jordan deformation of GL(3).

In [DMMZ] Manin formed the non-standard deformations of GL(2). One of them is the Jordan deformation, which was systematicaly studied in [EOW]. Here we give its analogue for G L ( 3 ) . We show that the new bialgebra is poincar6, too. Let

where A! is the dual of A,(see [MI). Let E = E ( S , A ) , then the matrix R is given by (qij = -qji,Pij = - p j i )

is Yang-Baxter and RZ = I , it is easy to show that S and A are compatible. Let us define the ordering in E as follows, Z: < 2: iff i > k or i = k,j > 1. Then

we have

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POINCA& SERIES OF QUANTUM MATRIX BIALGEBRAS 889

Theorem 4.3 E is poincare' and the normally ordered monomials with respect to the given order form a basis of E.

Proof. A direct computation shows that normally ordered monomials span E. By applying (3.6) we get the result. Q.E.D.

ACKNOWLEDGEMENTS

The author likes to thank Professor Yu.Manin for pointing him t o these problems, Professor J.Wess, who makes it possible that this work was written in his institute and Professor B.Pareigis, who gave him very many useful advices.

References

[ATS] Artin, Schelters, Tate. Quantum Deformation of GL(n) . Comm. on Pure and Applied Math. Vol. XLIV, 879-895(1991).

[B] Bergman. The Diamond Lemma For Ring Theory. Advances in Math. 29, (1978), (178-218).

[Bo] A.Bonda1 Non-commutative deformations and Poinson brackets on projective spaces. Preprint MPI/93-67.

[CG] E.Cremer, J.-L.Gervais. The Quantum Groups Structure Associated With Non-linearly Extended Virasoro Algebras. Comm. Math. Phys. 134(1990)619- 632.

[DMMZ] E.Demidov, Yu.Manin, E.Mukhin, D.Zhdanovich. Non-standard Deforma- tions of GL(n) and Constant Solutions of the Yang-Baxter Equation. Progr. Theor. Phys. Suplement. 102(1990) 203-218.

[EOW] H.Ewen, O.Ogievetsky, J.Wess. Quantum Matrices in Two Dimensions. Pre- print MPI-PAE/PTh 18/91.

[FRT] L. Faddeev, N. Reshetikhin. L. Takhtajian. Quantisation of Lie Groups and Lie Algebras. Algebra i analys. 1:1(1989), 178.(Leningrad Math. J. 1,193(1990).

[GI Gurevich. Hecke Symmetry and Quantum Determinants. Soviet Math. Iloklad Vol. 38(1989) 555-559.

[MI] Yu.I.Manin. Multiparametric Quantum Deformation of the General Linear Su- pergroups. Corn. Math. Phys. 123,163-175(1989).

[M2] Yu.I.Manin. Quantum groups and Non-commutative Geometry. GRM, Univ. de Montreal, 1988.

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[Sl] A.Sudbery. Consistent Multiparameter Quantisation GL(n). J. Phys A,(23), (1990), 697-704.

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Received: February 1994

Revised: July 1994

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