IC/93/157 / {•'• i INTERNATIONAL CENTRE FOR

\ ':'••• \ li /-W THEORETICAL PHYSICS

FINITE-DIMENSIONAL REPRESENTATIONS

OF THE QUANTUM SUPERALGEBRA Uq[gl{n/m)] AND RELATED ^-IDENTITIES

T.D. Palev INTERNATIONAL N.I. Stoilova ATOMIC ENERGY AGENCY and

J. Van der Jeugt UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION MIRAMARE-TRIESTE

IC/93/157 ABSTRACT

Internationa! Atomic Energy Agency Explicit expressions for the generators of the quantum Mipcralgrliirt U^\gi{n jut)) acting and on a class of irreducible representations are given, The class under consideration consists United Nations Educational Scientific and Cultural Organization of all essentially typical representations: for these be satisfied is shown to reduce to a set of ^-number identities.

FINITE-DIMENSIONAL REPRESENTATIONS OF THE QUANTUM SUPERALGEBRA Uq[gl(n/m)} AND RELATED ^-IDENTITIES

T.I). Palr-v ' International Centre for Theoretical Physics, Trieste, Italy, Department of Applied Mathematics and Computer Stience, University of Ghent, Krijgslaan 281 S9, H UOOO Ghent, Belgium and Institute for Nuclear Research and Nuclear Energy, 1784 Sofia, Bulgaria, 2

N.I. Stoilova Department of Applied Mathematics and Computer Science, University of Ghent, Krijgslaan 281 S9, B-9000 Ghent, Belgium and Institute for Nuclear Research and Nuclear Energy, 1784 Sofia, Bulgaria

J. Van der Jeugt 3 Department of Applied Mathematics and Computer Science, University of Ghent, Krijgslaan 281-S9, B-9000 Ghent, Belgium.

M1RAMARE-TRIESTE

June 1993

'E-mail: [email protected]. 'Permanent address. 'Research Associate of the N.F.W.O. (National Fund for Scientific Research of Bel- gium). E-mail: [email protected]. tht relations were proved in these representations Some of the relations acitially reduce to interesting ^-number identities, which can be proved using the Residue theorem of complex analysis- We conclude the paper with some comments and further outlook.

2 The Lie superalgebra gl(njm) and Gel'fand-Zetlin patterns

The Lie superalgebra G — j/(n/m) can be defined [Ifi, 28] througli its natural matrix realization

1 Introduction H gl{n/m)^{T- ( ^ n ]|.4fAlrIniJJt:WB,,1,('t.«,,l,,,l)€.W,,,«), (i) This paper is devoted to the study of a class of finite-dimensional irreducible representations of the quantum superalgebra (•',[gl(n/m)\ The main goa! is to present explicit actions of the Uj[gt(n/m)] where Afpl, is the space of all p x i/ complex matrices. The even suhalgehra i//(>i/>n)n has if = 0 and generating elements acting on a Gel'fand-Zetlin-like basis, and to discuss some of the

tions related to the Yang-Baxter equation [15], have now become an important and widely used concept where (t,£* £ {0,1} = Z3 If algebra G is a deformation of the universal enveloping algebra of G endowed with a space of matrices [ 1 and by y((n/in)_j the space of iiialrn.-s f j. Then G - gl(n/m)

Hopf algebra structure. The first example was given in [19, 30], and soon followed the generalization to H T 1 has a Z-grading which is consistent with the Zj-grading [28], namely G — G^x H ^ i^|iC + i with G ^ = GQ any Kac-Moody Lie algehra with syiumtriaable Cartan matrix [4, 12], For the deformation of the envelop- and f.'i = (7_i iji G+i Note that gl[n/in)a = gl(n) €• gl('n) Kor elements r uf ijt(n/m) given by (1), ing algebra of a Lie superalgebra we mention the case of osp(l/2) (20, 21], later to be extended to Lie one defines the supertrace as str(i) = tr(/l) - ir(D) The Lie supi-ralgthra ijl(njm) is not simple, and superalgebras with a symmetmable Cartan matrix [32] including the basic [16] Lie superalgebras [1, 2]. (for n ^ in) one can define the simple superalgehra JS/(II/HI) as llie suliHlgebra consisting of elements Representations of quantum algebras have been studied extensively, particularly for generic ^-values with supertrare 0 However, the representation theory atgHnjm) or r.!{n/'tii} is iwiitially the same (the (i.e. q not a root of unity). In this case, finite-dimensional irreducible representations of sl{n) can situation is similar as for the classical Lie algebras gt(ri) and ,s/(?i))- and lirnn- we prefer to work with be deformed into irreducible representations of f/j[$/(n)] [13], and it was shown that one obtains all h gt(n/m) and in the following section with its Hopf superalgehra deforuial l"ii i ^[///(ii/in)]. finite-dimensional irreducible modules of r 9[s/(n)] in this way [27]. In [14]. explicit expressions for the A basis for G — gl(n/m) consists of matrices Ei (),J —].'l. . r = in i- ") willi en! ry 1 at position generators of If,[sl(>0] acting on the "undeformed" Gel'fand-Zetlin basis were given. It is in the spirit of } (i, j) and D elsewhere, A Cartan subalgphra // of G is spanned by the elements hj TT EJ, (j — 1,2,..., r), this work that our present paper should be seen. Here, we study a class of irreducible representations of and a set of generators of gl{n/m) is given by the hj (j = 1.. .., r) ami tin- ilc-ments e, = £i,i+i and tlie quantum superalgebra (',[j/(n/m)]. The class consists of the so-calied essentially lypical representations: /, = £,+i,i {i = 1,.. .,r - I). The space dual to // is //' and is described by 1 li<' forms a (i = 1,. . ., r) one can interpret these as irreducible representations for which a Gel'fand-ZetJin basis can be given. Just where < for n + 1 ,?< T. following the study of [22, 23]; a generic example, t',[j/(3/2)], was treated in [26]; and the induced module construction of Kac [17J was generalized to f/,[j/(n/m)] [35]. On the other hand, oscillator representations where 8^ is the Kronecker-6. The components of an element A 6 H' will he written as [m] = m ( ar) have been constructed (2,3,7,9] not only for U,\gl(n/m)] but also for other quantum superalgebras. [mjn mjr, . , . , mrr\ where A = Yl^i ir i d m,v are complex numbers. The elements of H' are The structure of the present paper is as follows. In Section 2 we recall the definition of the Lie called the weights. The roots of gl(n/m) are the non-zero weights of the adjoint n /resentation, and take superalgebra gl{n/m) and fix the notation. We also remind the reader of some representation theory of the form (, — f} (i ^ j) in this basis; the positive roots are those with 1 < i < j < r, and of importance gi{n/m) which will be needed in the case of U^[gl[n/jn)], in particular of the concept of typical, atypical, are the nm odd positive roots and essentially typical representations. For the last class of representations, the Gel'fand-Zetlin basis introduced in [24] is written in explicit form. In the next section, we briefly recall the definition of the ftp = U - ' with 1 < i < n and n + I < p < r. (4)

quantum superalgebra Uf[gl(n/m)], Section A contains our main results. We present the actions of the For an element A £ H' with components [m], the Kac-Dynkin labels (a ,,. .,n _i;a ;a i,.. .,a -i) ^MWl"/"1)] generators on the Gel*fand-Zetlin basis introduced, and we give some indications of how t n n n + r are given by Oj = rriir — m,+ ir for ; ^ n and «„ = mnr + mn+! r. Hence. A with components [m] will be called an integral dominant weight if m,r - m, + 1,r € Z+ = {0,1,2, ..} for all i ^ n (1 < i < r - 1). chain of subalgebrae gl{n/m) D gt{n/rn- 1) 0 • o gl(n/\) 3 gl{n) D gl(n- 1} D • • • 3 gl{l). However, For every integral dominant weight A = [m] we denote by v">(A) the simple Go module with highest in order to be able to define appropriate actions of the »»-<> quantities (ir. a module with highest weight A = [m] is essentially typical if and only if where U(G-i) is the universal enveloping algebra of Ci- It follows that dimV(A) = 2"m dim V°(A).

By definition, F(A) is a highest weight module; unfortunately, V{\) is not always a simple G module. It {lir.l* Lr] (I {!„+,,,, !„+,,,. + 1 , /,. + 1,r + 2, . . , lrr } = 0. (8) contains a unique maximal (proper) submodule Af(A), and the quotient module The explicit form of the action [23, 24] will not be repeated liere, but tin- render interested can deduce (6) it from relations (24-30) of the present paper by taking the limit q — I (in far*. the limit of our present relations also improve some minor misprints in the transformations of tht- C \'I, basis as given in [23, 24}) is a finite-dimensional simple module with highest weight A. In fact, Kac [17] proved the following : It is necessary, however, to recall the labelling of the basis vectors for thcsi' modules, since the labelling of basis vectors of representations of the quantum algebra f',[i[/(ri/iit|| is ••x.ully the same (note that also Theorem 1 Every finite-dimensional simple G module is tsomorphic to a module of type (6), inhere A = for the quantum algebra t' [j/(n)], the finite-dimensional repriwiilaiiniis ran be labelled by the same [m] is integral dominant. Moreover, eiierji finite-dimtnstonai simple G module is uniquely characterized ( Gel'fand-Zetlin patterns as in the non-deformed case o\ g\\n), when i/ is not. a root of unity [14]). by its integral dominant highest wetght A. Let [in] be the labels of an integral dominant weight A. Associated wit li [rrj] we define a pattern |*n)

An Integra! dominant weight A = [m] (resp, F(A), resp. V(A» is called a typical weight (reep. a typical of r(r + l)/2 complex numbers mtl (1 < i < j < r) ordered as m Hie usual (icl'fand-Zetlin basis for

Kac module, reap, a typical simple module) if and only if (A + p\0ip) # 0 for all odd positive roots li,p of (4), where 2p is the sum of all positive roots of G. Otherwise A, F(A) and V(A) are called atypical. The importance of these definitions follows from another theorem of Kac [17] :

Theorem 2 The Kite-module F(A) is a simple G module if and only if A is typical

For an integral dominant highest weight A = [m] it is convenient to introduce the following labels [24] :

l,r-m,r-i + n+l, (l

In terms of these, one can deduce that {fi + p\P,p) = J;r-'pr, and hence the conditions for typicality take Such a pattern should satisfy the following set of conditions

a simple form. 1. the labels rn,r of A are fixed for all patterns, For typical modtles or representations one can say that they are well understood, and a character 2. m,p - rn,,,., = S.^-j £ (0, 1), (I < i < n- n + I < p < r), formula was given by Kac [17]. A character formula for all atypical modules has not been proven so U. mlt> — mi+1 p 6 Z+, (1 < i < n — 1; ri + 1 < p < r}n 4. 7n( j + , — m«j £ Z+ and m, j - i)i1 + i J + i £ Z+, (1 < i < j" < ti — 1 or n + 1 < i < j < r — 1). far, but there are several breakthroughs in this area : for singly atypical modules (for which the highest (10) weight A is atypical with respect to one single odd root ft ) a formula has been constructed [34]; for all p The last condition corresponds to the in-betweenness condition and ensures that the triangular pattern atypical modules a formula has been conjectured [33]; for atypical Kit-modules the composition series to the right of the m x n rectangle 7n;p (1 < i < n; n + 1 < p < rj in (H) corresponds to a classical has been conjectured [11] and partially shown to be correct [31]. On the other hand, the modules for GelTand-Zetlin pattern for gl{m), and that the triangular pattern below this rectangle corresponds to a which an explicit action of generators on basis vectors can be given, similar to the action of generators of Gel'fand-Zetlin pattern for gl{n). gl(n) on basis vectors with Gel'fand-Zetlin labels, is only a subclass of Ihe typical modules, namely the The following theorem was proved [24] : so-called essentially typical modules [24], the definition of which shall be recalled here. For simple gi(n) modules the Gel'fand-Zetlin basis vectorii [10] and their labels - with the conditions Theorem 3 Let A = [m] be an essentially typical highest weight. Then the set of all patterns (9) ("in-betweenness conditions") - are reflecting the decomposition of the module with respect to the chain of satisfying (10) constitute a basis for the (typical) Kac-module V(\) — V(A). subalgebras gl{n) D gt(n-\) D • • • D gl(l) In trying to construct a similar basis for the finite-dimensional The patterns (9) are referred to as Gel'fand-ZetEin (GZ) basis vectors for V(A) and an explicit action of modules of the Lie superalgebra gl(nfm) it was natural to consider the decomposition with respect to the the gl(n/m) generators h, (1 < j < r), e; and fi (1 < i < r — 1) has been given in Ilef. [24]. In the following section we shall recall the definition of the quantum algebra U,\gl(n/mj\. We shall 4 The £/?[)±ij is the pattern obtained from \n\) by replacing the entry m,j by m,j ± 1. T r 1 The quantum superalgebra V, = U,,[gl[n/m)] is the free associative superalgebra over C with parameter The following is the main result of this paper (as usual, [r] stands for (q - ij~ )/(q — q~ )) ; q € C and generators kj, k'1 (j = 1,2 r = n + m) and f;, /, (i = 1,2, ...,r - 1) subject to the Theorem 4 For generic values of q rvery essentially typical gl(n/m) iwlilf l'(A) with hight.il weight following relations (unless stated otherwise, the indices below run over all possible values) : A can it deformed into on irreducible 0',\gl(n/m)] module W(A) with (*r 5auir underlying vector space The Cartan-Kac relations : and with the action of the generators gtvtn by :

(11) k,\m) = ), (1 < t < r). (24) ,; (12)

The Serre relations for the e< (t-Serre relations) : l»')in, (27) 4 = 0, (16)

<-( + t«f = 0, for i >E • .,n + m-2}; (17)

+ Cje?+1 =0, for i , TJ + TO — 2}; (18) (28) :n-i«n+ieB = 0; (19)

• The relations obtained from (16-19) by replacing every e; by /; (/-Serre relations).

Equation (19) is the so-called extra Serre relation [8, 18, 29]. The Zj-grading in t/, is defined by the requirement that the only odd generators are en and /„; the degree of a homogeneous element a of Ut shall be denoted by deg(o), It can be shown that t/, is a Hopf superalgebra with counit c, comultiplication A and antipode S, defined by : f^ / n;::t,p,,-.-',,mn:::t,['»« e(«0 = *(/0 = 0, f(ij) = l; (20)

i(«() = e, if • T (29)

A(en) = e (21)

S(kj) = kj1, l S[ei) = -qet, S(fi) = -q~ fi, if t # n, (22) S(t») = -en, S(/n) = -/n. Remember that A : (/, -. U, ® V, is a morphism of graded algebras, and that the multiplication in

Uq ® U^ is given by

6 and hence the identity (33) holds. For the other cases, the method of proof is similar and we shall no longer iiimiiuii I hi- details. For (14) with i = h < n — 1, the identity to be verified is of the following type : i + l

n!cans tnat takes a values from l0 n w th lf a In the above expressions, J^,,^, or n*,«. = i * " ' ' * ^ '• + rj - a , (35) vector from the rhs of (24-30] does not belong to the module under consideration, then the corresponding term is lero even if the coefficient in front is undefined; if an equal number of factors in numerator and or, using a similar transformation as before, denominator are simultaneously equal to zero, they should be cancelled out. Eqs. (25,26) are the 1 same as in [14]; they describe the transformation of the basis under the action of the gl(n) generators. To conclude this section, we shall m.ike a number of comments on the proof of this Theorem- Provided A.YI^M,~'h)n;=l(.-I,-^A, that all coefficients in (24--30) are well denned (which is indeed the case under the conditions required here), it is sufficient to show that the explicit actions (24-30) satisfy the relation (11-19) (plus, of course, also the /-Serre relations). The irredueibility then follows from the results of Zhang [35] or from the observation that for generic q a deformed matrix element in the GZ basis is zero only if the corresponding This identity is proven by taking the function /(?) = f]j(- - "j)(; " r/'f j)/(- Ilj £ - - Ai)(i - V2/lj)) non-deformed matrix element vanishes, and applying tlie same Residue theorem, Finally, the ltiost coniphcatc-d < :w: is (I 1 j with i = i> > ;i. The To show that (II), (12) and (13) are satisfied is a straighforward matter. The difficult Cartan-Kac identity to prove is (with s = p — n) : relations to be verified are (!4) and (15). We shall consider one case in more detail, namely (15). This relation, with the actions (24-30), is vaiid if and only if ^ h-/* + ! (in + l —

*," + ! - (M-l) + '-." + 1 - 'n+ (31)

Putting a, = (,.„+] for i - 1,2 n, 6f = (* n_i for i = 1, 2, .., n - 1 and *„ = /„+!„ + !. the identity between ij-numbers to be proved reduces to (37)

(32) Herein, the /t are equal to S( p_i — Slp. anil since the 9'a take only the valuta (I ami I, the /t"s take only the values 0,±l, To prove (37), one again has to use the same l.cliiiii|iic on a fund ion Using the explicit definition of a ^-number, anci relabelling q2"' = A,, and

'(-) = • V"^ Fit = ]{A.i — B]c) _ B\By • • • Bn (33)

However, it turns out thai (37) is true as a general identity only when in the third summation the factor To prove this last identity, consider the complex function (/i) is replaced by [/t]. In the present case, this can be done without harm since for the values i = 0, ±1 we have indeed that [i] = x. This completes the verification of the Cartan-Kac relations. (34) For the e-Serre relations, the calculations are extremely lengthy, but when collecting terms with the same Gel'fand-ZetSin basis vector and then taking apart the common factors, the remaining factor always This function is hotomorphic over C except in its singular poles 0, A\,...,An (under the present condi- reduces to a simple finite expression which is easily verified to be zero These expressions always reduce tions, all At are indeed distinct). Let C be a closed curve whose interior contains all these poles. Then to one of the following (trivia!) identities : the Residue Theorem of complex analysis implies that §cf{z)dt = 2Tri(Res(0) + 2r=i R«(A))- It is easy to see that Res(0) = lim,-o/(*)* = (Bi "' Bn)/(Al • An) and that Rea[A.) = lim,-,,, f{z)[z - A<) = (38) nU=i(^f - B )/(Ai IlMi=i(^ - **))• On the other hand' k (39)

/ f(z)dz - -2)riRes(oo) = -2JT« lim [~z)J{z) = 2«, [21 (40) Jc "~a> 8i.B1JiK.-iJ?- -

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12 ••-» ••••'• at