Multiparameter Quantum Supergroups Michiel Hazewinkel
[email protected] CWI P.O. Box 94079, 1090 GB Amsterdam, The Netherlands
Abstract
This paper is supplementary to my paper \Multiparameter Quantum Groups and Multiparameter R-Matrices", [5]. Its main purpose is?to point out that among the single block solutions of the Yang-Baxter equation given in [5] there occurs an n+2m + 1 parameter quantum deformation of the supergroup GL(mjn) for every n; m 1.
AMS Subject Classi cation (1991): 16W30, 16W55, 17B70, 17C70, 81R50, 17B37 Keywords & Phrases: supergroup, quantum supergroup, Yang-Baxter equation, multiparameter quantum algebra, Hopf algebra, multiparameter quantum supergroup, bialgebra, FRT construction.
1. Introduction
Let K be any eld of charcteristic 6= 2 and consider the free associative algebra K < t > = K < t11 ; : : : ; t1n ; t21 ; : : : ; t2n ; : : : ; tn1 ; : : : ; tnn > over K in n2 (noncommuting) indeterminates t11 ; : : : ; tnn . ab ) be any n2 n2 matrix with coecients in K (and its rows and columns numLet R = (Rcd bered 11; : : :; 1n; 21 2n; : : : ; n1; : : : ; nn with lower indices for columns and upper indices for rows). Consider the n2 n2 elements making up the n2 n2 matrix RT1 T2 ? T2 T1R (1.1) where
0 t t 1 n CC B B CC ; T B . .. T =B CA B @ 1 1
1
1
= T I n ; T2 = I n T
(1.2)
tn1 tnn
Let I (R) be the two-sided ideal in K < t > generated by the elements of (1.1). On K < t > consider the standard \matrix function algebra" comultiplication
tij 7! tik tkj
(1.3) where the Einstein summation convention is in force, i.e. on the righthand side of (1.3) a sum over k is implied. This makes K < t > a bialgebra and it is always the case that I (R) is a bialgebra ideal; cf e.g. [4] for a proof. Thus Kn (R) = K < t > =I (R) (1.4) inherits a bialgebra structure. This is sometimes called the FRT construction (Faddeev - Reshetikin - Taktadzhyan). It was also indepedently noticed by other authors. Relations like (1.1) rst came up in the work of the Faddeev group in what was then called Leningrad (now Sankt Petersburg) in the
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context of quantum completely integrable dynamical systesm, [3]. In that context they were called \Fundamental Commutation Relations" and they form the basis for the socalled \Algebraic Bethe Ansatz" approach to nding the eigenvalues and vectors of quantum Hamiltonians. For certain, by now well known, solutions R of the (quantum) Yang-Baxter equation R12 R13 R23 = R23 R13 R12 (1.5) the bialgebra Hn (R), localized at a quantum determinant, are the Hopf algebras of functions of the one parameter \standard" quantum groups such as GLq (n). E.g. for n = 2; GLq (2) (over K ) is obtained from the R-matrix 0q 0 0 01
B B 0 B B B B B @0
CC CC C 0C CA
1 q ? q?1 0 0
1
(1.6)
0 0 0 q In [5] all solutions of the Yang-Baxter equation (1.5) were determined that satisfy the additional condition ab = 0 unless fa; bg = fc; dg Rcd (1.7) i.e. an entry of R is zero unless its sets of upper and lower indices are equal. The example (1.6) satis es (1.7). Among the solutions of the Yang-Baxter equation (1.5) given in [5] are what are there called \single block cell size 1" solutions. These look as follows \Single block cell size 1" solutions. Choose two elements y; z 2 K; z 6= 0. For each i > j ; i; j 2 f1; : : : ng choose 0 6= xij 2 K . Consider the quadratic scalar equation X 2 = yX + z (1.8) with solutions 1 ; 2 (assumed to exist in K ). Now de ne an n2 n2 matrix R over K as follows Riiii = 1 or 2 ; i = 1; : : : ; n
Rjiij = y
if i < j ; i; j 2 f1; : : :; ng
Rjiij = 0
if i > j ; i; j 2 f1; : : :; ng
Rijij = xij
if i > j ; i; j 2 f1; : : :; ng
Rijij = zx?ji1
(1.9)
if i > j ; i; j 2 f1; : : :; ng Then R is a solution of the Yang-Baxter equation (1.5). For instance the example (1.6) arises by taking y = q ? q?1 ; z = 1 (so that the solutions of (1.8) are 11 22 1 = q; 2 = ?q?1); x21 = 1, and taking R11 = R22 = q. ? The \single block cell size 1" solutions described above form 2n n2 + 2 parameter families indexed by the possible choices 1 or 2 which intersect in points where y and z are such that 1 = p 2 . Multiplying R with a nonzero scalar in K does not change the ideal I (R). For instance if z? exists in K , the parameter z can be normalized to 1. Thus these solutions give rise to a number of n2 + 1 ? parameter families of \quantum groups". Among these are n2 + 1 parameter quantum deformations of the supergroups GL(ljm); l + m = n; m 1. These I now proceed to describe.
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?n + 1 parameter quantum deformation of GL(ljm); l + m = n.
In the \single block" recipe above take z = 1, so that 1 = q; 2 = q?1 . Take Riiii = q for ab as in (1.9). The supergroup 1 i l; Riiii = ?q?1 for l + 1 i l + m, and the remaining Rcd GL(ljm) itself sits at the spot where q = 1; xij = 1 for i l or j l and xij = ?1 for i; j l + 1. More precisely the corresponding Hn (R) describes the bialgebras of functions for the matrix algebras of functions from which the corresponding quantum supergroups arise by localization at the appropriate quantum super determinant. For instance for l = 1; m = 2, the R-matrix for GL(1j2) is given by 1 01 CC B 1 B CC B 1 B 2
B B B B B B B B B B B B @
1
?1
?1 1
?1
?1
CC CC CC CC CC CC A
(1.10)
? and its 4 = + 1 parameter quantum deformation is given by the R-matrix 0q B x? q ? q? B ? B x q ? q? B B B B x B B ?q? B B x? q ? q? B B B B x B @ x 3 2
1 21
1
1 31
1
21
1
1 32
1
31
32
?q?
1
1 CC CC CC CC CC CC CC CC A
(from which (1.10) ? arises by taking q = 1; x21 = x31 = 1; x32 = ?1). ? 1 parameter deforNote that the n2 + 1 parameter quantum deformation of GL(n) and the n2 + p mation of GLjl)m); l + m = n intersect at the points where q = ?q?1 = i = ?1. 2. Algebra of functions of super space
Consider a superspace V = V0 V1 over the eld K of dimension ljm so that the summand V0 of even vectors is of dimension l and the space V1 of odd vectors is of dimension m. The algebra of (algebraic) functions of V is, [1,2,6], and especially [7], Aljm = K [X 1; : : : ; X l] K < l+1 ; : : : ; l+m > (2.1) where the X 1; : : : ; X l are commuting indeterminates, X i X j = X j X i , and the l+1 ; : : : ;l+m are anticommuting indeterminates, l+1 l+j = ? l+j l+1 (Grassmann indeterminates). In other words Aljm is the quotient of the free associative algebra K < Z 1; : : : ; Z l+m > subject to the relations Z i Z j = Z j Z i if i or j l and Z i Z j = ?Z j Z i if both i and j are l + 1. 3. Algebra of functions on supermatrices
One way to nd the algebra of functions for supermatrices is to consider the free algebra K < t > and K < Z > and the standard left matrix coaction
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Z i 7! tij Z j
(3.1) Now consider the relations de ning Aljm as a quotient of K < Z > and calculate the relations that are needed between the tij in order that (3.1) respects the relations de ning Aljm . For instance from X 1 2 = 2 X 1 in A1j2 one gets, (t11 X 1 + t12 2 + t13 3 )(t21 X 1 + t22 2 + t23 3 ) =
= (t21 X 1 + t22 2 + t23 3 )(t11 X 1 + t12 2 + t13 3 ) which, using X 1 2 = 2 X 1 ; ( 2 )2 = ( 3 )2 = 0; 2 3 = ? 3 2 , gives t11 t21 = t21 t11
(3.2) (3.3)
t11 t22 + t12 t21 = t22 t11 + t21 t12 t11 t23 + t13 t21 = t21 t13 + t23 t11
(3.4)
t12 t23 ? t13 t22 = t22 t13 ? t23 t12
(3.6)
(3.5)
This gives precisely half of the relations needed. The other half comes from the right coaction on the dual superspace K [Y1 ; : : : ; Yl ] K < l+1 ; : : : ; l+m > coming from the right matrix coaction Zi 7! Zj tji (3.7) For instance for l = 1; m = 2, one of the relations coming from 2 3 = ?3 2 is t12 t23 + t22 t13 = ?t23 t12 ? t13 t22 (3.8) which combined with (3.6) gives (unless char(K ) = 2) t22 t13 = ?t13 t22 ; t12 t23 = ?t23 t12 (3.9) These calculations are very similar to those in section 2 of [5] and as in appendix 1 of [5] one veri es that the resulting relations generate a bialgebra ideal in K < t >. The nal result is, see also [7], 3.10 Theorem. The bialgebra of functions Aljmljm for the supermatrices of size (ljm) (ljm) over
K is the quotient of K < t > given by the relations tab trs = trs tbb if 1 a; b l
tab trs = trs tab if 1 a; s l; l + 1 b; r l + m tab trs = ?trs tab if 1 a l; l + 1 b; s l + m
(3.10)
tab trs = ?trs tab if 1 b l; l + 1 a; r +m tab trs = trs tab
if l + 1 a; b; r; s l + m
These relations are easily remembered by partitioning the matrix T = (tij ) in four parts as indicated
l
l m
m
0 I II 1 A=T @ III IV
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and assigning the bidegrees (0; 0); (1; 0); (0; 1); (1; 1) to the variables in the blocks I; II; III; IV respectively. These bidegrees correspond to even to even, odd to even, even to odd, odd to odd maps. Writing d1 and d2 for the rst and second components of the bidegree respectively, the commutation rules (3.11) become tab trs = (?1)d1(tab )d1(trs )+d2 (tab )d2 (trs ) trs tab (3.11) It is now straightforward to verify 3.13 Proposition. Let R be the diagonal n2 n2 matrix, n = l + m, with Rijij = 1 for i l or j l and Rijij = ?1 if i > l and j > l. Then the ideal I (R) is precisely that of the relations (3.11). I.e. Hn (R) for this particular R is the bialgebra of (algebraic) functions for (ljm ljm) supermatrices. 4. Multiparameter quantum supergroups
Now consider the R-matrtix already speci ed in the introduction, to obtain 4.1. Theorem. Let R be the n2 n2 matrix, n = l + m, with Rijij = q ? q?1 if i < j; Riiii = q if ab = 0 in all other case. Then i l; Riiii = ?q?1?if i > l; Rijij = xij if i > j; Rijij = x?ij1 if i < j; Rcd n the Hn (R) are an 2 + 1 parameter family of bialgebra deformations of the bialgebra of functions on ljm ljm supermatrices. This algebra itself is located at q = 1; xij = 1 for i l or j l, xij = ?1 if i; j l + 1. 5. Concluding remarks
The algebra of functions of the supergroup GL(ljm) arises from Aljmljm by localization at the superdeterminant (polynomial). For the single parameter quantum deformation of Aljmljm de ned by xij = 1 for i l or j l; xij = ? if i; j l + 1 there is a corresponding quantum superdeterminant, [7]. This should ? generalize to the multiparameter case, but that remains to be veri ed. For the n2 +1 parameter quantum deformation of GL(n) the Poincare series of each of the deformed algebras is the same as that of the function algebra of GL(n) itself. This is not the case for the quantum deformation of GL(ljm). At q = ?q?1, i.e. q = i, that is precisely the places where the two families can intersect, there is a jump. Over C the quantum groups (algebras of functions) and quantum super group (algebras of functions) with z = 1 and z = ?1 are isomorphic. It remains to sort out what is the situation over R where z = ?1 cannot be eaesily normalized away to z = 1. References [1] Claudio Bartozzi, Ugo Bruzzo, Daniel Hernandez-Ruiperez, The geometry of super-
manifolds, KAP, 1991. [2] F.A. Berezin, Introduction to superanalysis, Reidel, 1987. [3] L.D. Faddeev, N.Yu. Reshetikhim, L.A. Taktadzhyan, Quantized Lie groups and Lie algebras, Leningrad Math. J. 1 (1990), 491{513. [4] Michiel Hazewinkel, Introductory recommendations for the study of Hopf algebras in mathematics and physics, CWI Quarterly 4:1 (1991), 3{26. [5] Michiel Hazewinkel, Multiparameter quantum groups and multiparameter R-matrices, Acta Appl. Math. 41 (1995), 57{98. [6] D.A. Leites, Theory of supermanifolds. Dept. Math. Methods of Automization, Scienti c Studies and Prognosis, Karelian Dept. Akad. Nauk SSSR, Petrozavodsk, 1983, in Russian. [7] Yuri I. Manin, Topics in noncommutative geometry, Princeton Univ. Press, 1991.
