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 <i Cerfand Zellin basis is known. The INTERNATIONAL CENTRE EOR THEORETICAL PHYSICS verification of the quantum supe.raigebra relations U> 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 <j-number identities C = 0; the odd subspacf j((»/w}[ has .4 = (I and 1) = I), '['lie hrackrl is .I«I,TILII[H d by related to these representations [a,b] = ab - (-l)";'fcr!. V«: £ (.'„ and VI: C (,',,, (2) Quantum groups [S], funling their origin in the quantum inverse problem method [6] and in investiga- tions related to the Yang-Baxter equation [15], have now become an important and widely used concept where (t,£* £ {0,1} = Z3 If <i 6 d\. then (t = deg(n) is called ill-' ilegn-i' of u, and an element of T in various branches of physics and mathematics- A quantum (super Jalgebra Ut[G] associated with a G = 6 ^ ej^ G] is called homogeneous if it belongs to either G*n iar elw f^ Wr d-note by gt(n/i7i) + i the (simple) Lie (super)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 <j : r —• A for 1 < j < n and < : J — /^j for 1 < j < m. atnl «|irrc J- is ijivrii as in (1). On as in the case of s((n), the basis will remain undeformed, and the deformation will arise in the action of n I1+J //• there is a bilinear Form defined deduced From the suprrintie on (!. and tx[.liiil.lv given hy [34]: the quantum superalgebra generators on the basis vectors. In the literature, some results have already appeared for representations of quantum superalgebras of type V,[gl(nj-m)] : representations of r/,[ji(n/l)] (both typical and atypical] were examined in [25] ((•i'p) =0, for 1 < i < n and ii 4 1 < ]> < 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 <jl(n/m) generators on basis vectors with respect to this decomposition, it is necessary that at every step in this reduction the corresponding modules weight A; this is simply the finite-dimensional flj(n)8s'(m) module with gl{n) labels {mlr,.