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Finite-dimensional representations of the quantum superalgebra

arXiv:hep-th/9306149v1 28 Jun 1993Finite-dimensional representations of the quantum superalgebraUq[gl(n/m)] and related q-identitiesT.D. Palev∗,a,b,c), N.I.

Stoilovab,c) and J. Van der Jeugt†,b)a) International Centre for Theoretical Physics, 34100 Trieste, Italy.b) Department of Applied Mathematics and Computer Science, University of Ghent, Krijgslaan 281–S9,B-9000 Gent, Belgium.c) Institute for Nuclear Research and Nuclear Energy, 1784 Sofia, Bulgaria (permanent address).AbstractExplicit expressions for the generators of the quantum superalgebra Uq[gl(n/m)] actingon a class of irreducible representations are given.

The class under consideration consistsof all essentially typical representations : for these a Gel’fand-Zetlin basis is known. Theverification of the quantum superalgebra relations to be satisfied is shown to reduce to a setof q-number identities.MSC numbers: 16W30, 17B40, 81R50.———————————–∗) E-mail : palev@bgearn.bitnet.†) Research Associate of the N.F.W.O.

(National Fund for Scientific Research of Belgium). E-mail : joris@fwet.rug.ac.be.1

Finite-dimensional representations of Uq[gl(n/m)]1IntroductionThis paper is devoted to the study of a class of finite-dimensional irreducible representations of thequantum superalgebra Uq[gl(n/m)].The main goal is to present explicit actions of the Uq[gl(n/m)]generating elements acting on a Gel’fand-Zetlin-like basis, and to discuss some of the q-number identitiesrelated to these representations.Quantum groups [5], finding 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 conceptin various branches of physics and mathematics. A quantum (super)algebra Uq[G] associated with a(simple) Lie (super)algebra G is a deformation of the universal enveloping algebra of G endowed with aHopf algebra structure.

The first example was given in [19, 30], and soon followed the generalization toany Kac-Moody Lie algebra with symmtrizable Cartan matrix [4, 12]. For the deformation of the envelop-ing algebra of a Lie superalgebra we mention the case of osp(1/2) [20, 21], later to be extended to Liesuperalgebras with a symmetrizable Cartan matrix [32] including the basic [16] Lie superalgebras [1, 2].Representations of quantum algebras have been studied extensively, particularly for generic q-values(i.e.

q not a root of unity).In this case, finite-dimensional irreducible representations of sl(n) canbe deformed into irreducible representations of Uq[sl(n)] [13], and it was shown that one obtains allfinite-dimensional irreducible modules of Uq[sl(n)] in this way [27]. In [14], explicit expressions for thegenerators of Uq[sl(n)] acting on the “undeformed” Gel’fand-Zetlin basis were given.

It is in the spirit ofthis work that our present paper should be seen. Here, we study a class of irreducible representations ofthe quantum superalgebra Uq[gl(n/m)].

The class consists of so-called essentially typical representations;one can interpret these as irreducible representations for which a Gel’fand-Zetlin basis can be given. Justas in the case of sl(n), the basis will remain undeformed, and the deformation will arise in the action ofthe quantum superalgebra generators on the basis vectors.In the literature, some results have already appeared for representations of quantum superalgebrasof type Uq[gl(n/m)] : representations of Uq[gl(n/1)] (both typical and atypical) were examined in [25]following the study of [22, 23]; a generic example, Uq[gl(3/2)], was treated in [26]; and the induced moduleconstruction of Kac [17] was generalized to Uq[gl(n/m)] [35].

On the other hand, oscillator representationshave been constructed [2, 7, 9] not only for Uq[gl(n/m)] but also for other quantum superalgebras.The structure of the present paper is as follows.In Section 2 we recall the definition of the Liesuperalgebra gl(n/m) and fix the notation. We also remind the reader of some representation theory ofgl(n/m) which will be needed in the case of Uq[gl(n/m)], in particular of the concept of typical, atypical,and essentially typical representations.

For the last class of representations, the Gel’fand-Zetlin basisintroduced in [24] is written in explicit form. In the next section, we briefly recall the definition of thequantum superalgebra Uq[gl(n/m)].

Section 4 contains our main results. We present the actions of theUq[gl(n/m)] generators on the Gel’fand-Zetlin basis introduced, and we give some indications of how2

the relations were proved in these representations. Some of the relations actually reduce to interestingq-number identities, which can be proved using the Residue theorem of complex analysis.

We concludethe paper with some comments and further outlook.2The Lie superalgebra gl(n/m) and Gel’fand-Zetlin patternsThe Lie superalgebra G = gl(n/m) can be defined [16, 28] through its natural matrix realizationgl(n/m) = {x = ABCD|A ∈Mn×n, B ∈Mn×m, C ∈Mm×n, D ∈Mm×m},(1)where Mp×q is the space of all p × q complex matrices. The even subalgebra gl(n/m)¯0 has B = 0 andC = 0; the odd subspace gl(n/m)¯1 has A = 0 and D = 0.

The bracket is determined by[a, b] = ab −(−1)αβba,∀a ∈Gα and ∀b ∈Gβ,(2)where α, β ∈{¯0, ¯1} ≡Z2. If a ∈Gα then α = deg(a) is called the degree of a, and an element ofG = G¯0 ⊕G¯1 is called homogeneous if it belongs to either G¯0 or else G¯1.

We denote by gl(n/m)+1 thespace of matrices0B00and by gl(n/m)−1 the space of matrices00C0. Then G = gl(n/m)has a Z-grading which is consistent with the Z2-grading [28], namely G = G−1 ⊕G0 ⊕G+1 with G¯0 = G0and G¯1 = G−1 ⊕G+1.

Note that gl(n/m)0 = gl(n) ⊕gl(m). For elements x of gl(n/m) given by (1),one defines the supertrace as str(x) = tr(A) −tr(D).

The Lie superalgebra gl(n/m) is not simple, and(for n ̸= m) one can define the simple superalgebra sl(n/m) as the subalgebra consisting of elementswith supertrace 0. However, the representation theory of gl(n/m) or sl(n/m) is essentially the same (thesituation is similar as for the classical Lie algebras gl(n) and sl(n)), and hence we prefer to work withgl(n/m) and in the following section with its Hopf superalgebra deformation Uq[gl(n/m)].A basis for G = gl(n/m) consists of matrices Eij (i, j = 1, 2, .

. ., r ≡m + n) with entry 1 at position(i, j) and 0 elsewhere.

A Cartan subalgebra H of G is spanned by the elements hj = Ejj (j = 1, 2, . .

. , r),and a set of generators of gl(n/m) is given by the hj (j = 1, .

. .

, r) and the elements ei = Ei,i+1 andfi = Ei+1,i (i = 1, . .

. , r −1).

The space dual to H is H∗and is described by the forms ǫi (i = 1, . .

. , r)where ǫj : x →Ajj for 1 ≤j ≤n and ǫn+j : x →Djj for 1 ≤j ≤m, and where x is given as in (1).

OnH∗there is a bilinear form defined deduced from the supertrace on G, and explicitly given by [34]:⟨ǫi|ǫj⟩= δij,for 1 ≤i, j ≤n;⟨ǫi|ǫp⟩= 0,for 1 ≤i ≤n and n + 1 ≤p ≤r;⟨ǫp|ǫq⟩= −δpq,for n + 1 ≤p, q ≤r. (3)where δij is the Kronecker-δ.The components of an element Λ ∈H∗will be written as [m] =[m1r, m2r, .

. .

, mrr] where Λ = Pri=1 mirǫi and mir are complex numbers.The elements of H∗arecalled the weights. The roots of gl(n/m) are the non-zero weights of the adjoint representation, and takethe form ǫi −ǫj (i ̸= j) in this basis; the positive roots are those with 1 ≤i < j ≤r, and of importanceare the nm odd positive rootsβip = ǫi −ǫp,with 1 ≤i ≤n and n + 1 ≤p ≤r.

(4)For an element Λ ∈H∗with components [m], the Kac-Dynkin labels (a1, . .

. , an−1; an; an+1, .

. .

, ar−1)are given by ai = mir −mi+1,r for i ̸= n and an = mnr + mn+1,r. Hence, Λ with components [m] will3

be called an integral dominant weight if mir −mi+1,r ∈Z+ = {0, 1, 2, . .

.} for all i ̸= n (1 ≤i ≤r −1).For every integral dominant weight Λ ≡[m] we denote by V 0(Λ) the simple G0 module with highestweight Λ; this is simply the finite-dimensional gl(n)⊕gl(m) module with gl(n) labels {m1,r, .

. .

mnr} andwith gl(m) labels {mn+1,r, . .

. , mrr}.

The module V 0(Λ) can be extended to a G0 ⊕G+1 module by therequirement that G+1V 0(Λ) = 0. The induced G module V (Λ), first introduced by Kac [17] and usuallyreferred to as the Kac-module, is defined byV (Λ) = IndGG0⊕G+1V 0(Λ) ∼= U(G−1) ⊗V 0(Λ),(5)where U(G−1) is the universal enveloping algebra of G−1.

It follows that dim V (Λ) = 2nm dim V 0(Λ).By definition, V (Λ) is a highest weight module; unfortunately, V (Λ) is not always a simple G module. Itcontains a unique maximal (proper) submodule M(Λ), and the quotient moduleV (Λ) = V (Λ)/M(Λ)(6)is a finite-dimensional simple module with highest weight Λ.

In fact, Kac [17] proved the following :Theorem 1 Every finite-dimensional simple G module is isomorphic to a module of type (6), where Λ ≡[m] is integral dominant. Moreover, every finite-dimensional simple G module is uniquely characterizedby its integral dominant highest weight Λ.An integral dominant weight Λ = [m] (resp.

V (Λ), resp. V (Λ)) is called a typical weight (resp.

a typicalKac module, resp. a typical simple module) if and only if ⟨Λ + ρ|βip⟩̸= 0 for all odd positive roots βip of(4), where 2ρ is the sum of all positive roots of G. Otherwise Λ, V (Λ) and V (Λ) are called atypical.

Theimportance of these definitions follows from another theorem of Kac [17] :Theorem 2 The Kac-module V (Λ) is a simple G module if and only if Λ is typical.For an integral dominant highest weight Λ = [m] it is convenient to introduce the following labels [24] :lir = mir −i + n + 1,(1 ≤i ≤n);lpr = −mpr + p −n,(n + 1 ≤p ≤r). (7)In terms of these, one can deduce that ⟨Λ+ ρ|βip⟩= lir −lpr, and hence the conditions for typicality takea simple form.For typical modules or representations one can say that they are well understood, and a characterformula was given by Kac [17].

A character formula for all atypical modules has not been proven sofar, but there are several breakthroughs in this area : for singly atypical modules (for which the highestweight Λ is atypical with respect to one single odd root βip) a formula has been constructed [34]; for allatypical modules a formula has been conjectured [33]; for atypical Kac-modules the composition serieshas been conjectured [11] and partially shown to be correct [31]. On the other hand, the modules forwhich an explicit action of generators on basis vectors can be given, similar to the action of generators ofgl(n) on basis vectors with Gel’fand-Zetlin labels, is only a subclass of the typical modules, namely theso-called essentially typical modules [24], the definition of which shall be recalled here.For simple gl(n) modules the Gel’fand-Zetlin basis vectors [10] and their labels – with the conditions(“in-betweenness conditions”) – are reflecting the decomposition of the module with respect to the chain ofsubalgebras gl(n) ⊃gl(n−1) ⊃· · · ⊃gl(1).

In trying to construct a similar basis for the finite-dimensionalmodules of the Lie superalgebra gl(n/m) it was natural to consider the decomposition with respect to the4

chain of subalgebras gl(n/m) ⊃gl(n/m −1) ⊃· · · ⊃gl(n/1) ⊃gl(n) ⊃gl(n −1) ⊃· · · ⊃gl(1). However,in order to be able to define appropriate actions of the gl(n/m) generators on basis vectors with respectto this decomposition, it was necessary that at every step in this reduction the corresponding modulesare completely reducible with respect to the submodule under consideration.

A sufficient condition isthat for every step in the above reduction the modules are typical, i.e. a typical gl(n/m) module V mustdecompose into typical gl(n/m −1) modules, each of which must decompose into typical gl(n/m −2)modules etc.

Such modules are called essentially typical [24], and a Gel’fand-Zetlin-like basis can beconstructed with an action of the gl(m/n) generators. In terms of the above quantities lir, a module withhighest weight Λ ≡[m] is essentially typical if and only if{l1r, l2r, .

. .

, lnr} ∩{ln+1,r, ln+1,r + 1, ln+1,r + 2, . .

. , lrr} = ∅.

(8)The explicit form of the action [23, 24] will not be repeated here, but the reader interested can deduceit from relations (24–30) of the present paper by taking the limit q →1 (in fact, the limit of our presentrelations also improve some minor misprints in the transformations of the GZ basis as given in [23, 24]).It is necessary, however, to recall the labelling of the basis vectors for these modules, since the labellingof basis vectors of representations of the quantum algebra Uq[gl(n/m)] is exactly the same (note that alsofor the quantum algebra Uq[gl(n)], the finite-dimensional representations can be labelled by the sameGel’fand-Zetlin patterns as in the non-deformed case of gl(n), when q is not a root of unity [14]).Let [m] be the labels of an integral dominant weight Λ. Associated with [m] we define a pattern |m)of r(r + 1)/2 complex numbers mij (1 ≤i ≤j ≤r) ordered as in the usual Gel’fand-Zetlin basis forgl(r) :|m) =m1r· · ·mn−1,rmnrmn+1,r· · ·mr−1,rmrrm1,r−1· · ·mn−1,r−1mn,r−1mn+1,r−1· · ·mr−1,r−1..................m1,n+1· · ·mn−1,n+1mn,n+1mn+1,n+1m1n· · ·mn−1,nmnnm1,n−1· · ·mn−1,n−1......m11(9)Such a pattern should satisfy the following set of conditions :1.the labels mir of Λ are fixed for all patterns,2.mip −mi,p−1 ≡θi,p−1 ∈{0, 1},(1 ≤i ≤n; n + 1 ≤p ≤r),3.mip −mi+1,p ∈Z+,(1 ≤i ≤n −1; n + 1 ≤p ≤r),4.mi,j+1 −mij ∈Z+ and mi,j −mi+1,j+1 ∈Z+,(1 ≤i ≤j ≤n −1 or n + 1 ≤i ≤j ≤r −1).

(10)The last condition corresponds to the in-betweenness condition and ensures that the triangular patternto the right of the m × n rectangle mip (1 ≤i ≤n; n + 1 ≤p ≤r) in (9) corresponds to a classicalGel’fand-Zetlin pattern for gl(m), and that the triangular pattern below this rectangle corresponds to aGel’fand-Zetlin pattern for gl(n).The following theorem was proved [24] :Theorem 3 Let Λ ≡[m] be an essentially typical highest weight.Then the set of all patterns (9)satisfying (10) constitute a basis for the (typical) Kac-module V (Λ) = V (Λ).The patterns (9) are referred to as Gel’fand-Zetlin (GZ) basis vectors for V (Λ) and an explicit action ofthe gl(n/m) generators hj (1 ≤j ≤r), ei and fi (1 ≤i ≤r −1) has been given in Ref. [24].5

In the following section we shall recall the definition of the quantum algebra Uq[gl(n/m)]. We shallthen define an action of the quantum algebra generators on the Gel’fand-Zetlin basis vectors |m) intro-duced here.

In other words, just as for the finite-dimensional gl(n) modules, one can use the same basisvectors and only the action is deformed.3The quantum superalgebra Uq[gl(n/m)]The quantum superalgebra Uq ≡Uq[gl(n/m)] is the free associative superalgebra over C with parameterq ∈C and generators kj, k−1j(j = 1, 2, . .

., r ≡n + m) and ei, fi (i = 1, 2, . .

., r −1) subject to thefollowing relations (unless stated otherwise, the indices below run over all possible values) :• The Cartan-Kac relations :kikj = kjki,kik−1i= k−1iki = 1;(11)kiejk−1i= q(δij−δi,j+1)/2ej,kifjk−1i= q−(δij−δi,j+1)/2fj;(12)eifj −fjei=0 if i ̸= j;(13)eifi −fiei=(k2i k−2i+1 −k2i+1k−2i)/(q −q−1) if i ̸= n;(14)enfn + fnen=(k2nk2n+1 −k−2n k−2n+1)/(q −q−1);(15)• The Serre relations for the ei (e-Serre relations) :eiej = ejei if |i −j| ̸= 1;e2n = 0;(16)e2i ei+1 −(q + q−1)eiei+1ei + ei+1e2i = 0, for i ∈{1, . .

. , n −1} ∪{n + 1, .

. .

, n + m −2};(17)e2i+1ei −(q + q−1)ei+1eiei+1 + eie2i+1 = 0, for i ∈{1, . .

. , n −2} ∪{n, .

. .

, n + m −2};(18)enen−1enen+1+en−1enen+1en+enen+1enen−1+en+1enen−1en−(q+q−1)enen−1en+1en = 0; (19)• The relations obtained from (16–19) by replacing every ei by fi (f-Serre relations).Equation (19) is the so-called extra Serre relation [8, 18, 29]. The Z2-grading in Uq is defined by therequirement that the only odd generators are en and fn; the degree of a homogeneous element a of Uqshall be denoted by deg(a).

It can be shown that Uq is a Hopf superalgebra with counit ε, comultiplication∆and antipode S, defined by :ε(ei) = ε(fi) = 0,ε(kj) = 1;(20)∆(kj)=kj ⊗kj,∆(ei)=ei ⊗kik−1i+1 + k−1iki+1 ⊗ei,if i ̸= n,∆(en)=en ⊗knkn+1 + k−1n k−1n+1 ⊗en,∆(fi)=fi ⊗kik−1i+1 + k−1iki+1 ⊗fi,if i ̸= n,∆(fn)=fn ⊗knkn+1 + k−1n k−1n+1 ⊗fn;(21)S(kj)=k−1j ,S(ei)=−qei,S(fi) = −q−1fi,if i ̸= n,S(en)=−en,S(fn) = −fn. (22)Remember that ∆: Uq →Uq ⊗Uq is a morphism of graded algebras, and that the multiplication inUq ⊗Uq is given by(a ⊗b)(c ⊗d) = (−1)deg(b) deg(c)ac ⊗bd.

(23)6

4The Uq[gl(n/m)] representationsLet Λ ≡[m] be an essentially typical highest weight, and denote by W(Λ) the vector space spanned bythe basis vectors |m) of the form (9) satisfying the conditions (10). On this vector space, we shall definean action of the generators of Uq = Uq[gl(n/m)], thus turning W(Λ) into a Uq module.

For convenience,we introduce the following notations : lij = mij −i + n + 1 for 1 ≤i ≤n, lpj = −mpj + p −n forn + 1 ≤p ≤r, and |m)±ij is the pattern obtained from |m) by replacing the entry mij by mij ± 1.The following is the main result of this paper (as usual, [x] stands for (qx −q−x)/(q −q−1)) :Theorem 4 For generic values of q every essentially typical gl(n/m) module V (Λ) with highest weightΛ can be deformed into an irreducible Uq[gl(n/m)] module W(Λ) with the same underlying vector spaceand with the action of the generators given by :ki|m)=qPij=1 mji−Pi−1j=1 mj,i−1/2|m),(1 ≤i ≤r),(24)ek|m)=kXj=1 −Qk+1i=1 [li,k+1 −ljk] Qk−1i=1 [li,k−1 −ljk −1]Qki̸=j=1[lik −ljk][lik −ljk −1]!1/2|m)jk,(1 ≤k ≤n −1),(25)fk|m)=kXj=1 −Qk+1i=1 [li,k+1 −ljk + 1] Qk−1i=1 [li,k−1 −ljk]Qki̸=j=1[lik −ljk + 1][lik −ljk]!1/2|m)−jk,(1 ≤k ≤n −1),(26)en|m)=nXi=1θin(−1)i−1(−1)θ1n+...+θi−1,n Qn−1k=1[lk,n−1 −li,n+1]Qnk̸=i=1[lk,n+1 −li,n+1]!1/2|m)in,(27)fn|m)=nXi=1(1 −θin)(−1)i−1(−1)θ1n+...+θi−1,n[li,n+1 −ln+1,n+1]× Qn−1k=1[lk,n−1 −li,n+1]Qnk̸=i=1[lk,n+1 −li,n+1]!1/2|m)−in,(28)ep|m)=nXi=1θip(−1)θ1p+...+θi−1,p+θi+1,p−1+...+θn,p−1(1 −θi,p−1)×nYk̸=i=1[li,p+1 −lkp][li,p+1 −lkp −1][li,p+1 −lk,p+1][li,p+1 −lk,p−1 −1]1/2|m)ip+pXs=n+1 −Qp−1q=n+1[lq,p−1 −lsp + 1] Qp+1q=n+1[lq,p+1 −lsp]Qpq̸=s=n+1[lqp −lsp][lqp −lsp + 1]!1/2×nYk=1[lkp −lsp][lkp −lsp + 1][lk,p+1 −lsp][lk,p−1 −lsp + 1]|m)sp,(n + 1 ≤p ≤r −1),(29)fp|m)=nXi=1θi,p−1(−1)θ1p+...+θi−1,p+θi+1,p−1+...+θn,p−1(1 −θip)×nYk̸=i=1[li,p+1 −lkp][li,p+1 −lkp −1][li,p+1 −lk,p+1][li,p+1 −lk,p−1 −1]1/2×Qp−1q=n+1[li,p+1 −lq,p−1 −1] Qp+1q=n+1[li,p+1 −lq,p+1]Qpq=n+1[li,p+1 −lqp −1][li,p+1 −lqp]|m)−ip7

+pXs=n+1 −Qp−1q=n+1[lq,p−1 −lsp] Qp+1q=n+1[lq,p+1 −lsp −1]Qpq̸=s=n+1[lqp −lsp −1][lqp −lsp]!1/2|m)−sp,(n + 1 ≤p ≤r −1). (30)In the above expressions, Pnk̸=i=1 or Qnk̸=i=1 means that k takes all values from 1 to n with k ̸= i.

If avector from the rhs of (24–30) does not belong to the module under consideration, then the correspondingterm is zero even if the coefficient in front is undefined; if an equal number of factors in numerator anddenominator are simultaneously equal to zero, they should be cancelled out. The eqs.

(25,26) are thesame as in [14]; they desribe the transformation of the basis under the action of the gl(n) generators.To conclude this section, we shall make a number of comments on the proof of this Theorem. Providedthat all coefficients in (24–30) are well defined (which is indeed the case under the conditions requiredhere), it is sufficient to show that the explicit actions (24–30) satisfy the relation (11–19) (plus, of course,also the f-Serre relations).

The irreducibility then follows from the results of Zhang [35] or from theobservation that for generic q a deformed matrix element in the GZ basis is zero only if the correspondingnon-deformed matrix element vanishes.To show that (11), (12) and (13) are satisfied is a straighforward matter. The difficult Cartan-Kacrelations to be verified are (14) and (15).

We shall consider one case in more detail, namely (15). Thisrelation, with the actions (24–30), is valid if and only ifnXi=1[li,n+1 −ln+1,n+1]Qn−1k=1[lk,n−1 −li,n+1]Qnk̸=i=1[lk,n+1 −li,n+1]="n−1Xk=1(lk,n+1 −lk,n−1) + ln,n+1 −ln+1,n+1#.

(31)Putting ai = li,n+1 for i = 1, 2, . .

. , n, bi = li,n−1 for i = 1, 2, .

. .

, n −1 and bn = ln+1,n+1, the identitybetween q-numbers to be proved reduces tonXi=1Qnk=1[ai −bk]Qnk̸=i=1[ai −ak] =" nXk=1(ak −bk)#. (32)Using the explicit definition of a q-number, and relabelling q2ai = Ai and q2bi = Bi, this becomesnXi=1Qnk=1(Ai −Bk)AiQnk̸=i=1(Ai −Ak) = 1 −B1B2 · · · BnA1A2 · · · An.

(33)To prove this last identity, consider the complex functionf(z) =Qnk=1(z −Bk)z Qnk=1(z −Ak). (34)This function is holomorphic over C except in its singular poles 0, A1, .

. .

, An (under the present condi-tions, all Ak are indeed distinct). Let C be a closed curve whose interior contains all these poles.

Thenthe Residue Theorem of complex analysis implies thatHC f(z)dz = 2πi(Res(0)+Pni=1 Res(Ai)). It is easyto see that Res(0) = limz→0 f(z)z = (B1 · · · Bn)/(A1 · · · An) and that Res(Ai) = limz→Ai f(z)(z −Ai) =Qnk=1(Ai −Bk)/(AiQnk̸=i=1(Ai −Ak)).

On the other hand,ICf(z)dz = −2πi Res(∞) = −2πi limz→∞(−z)f(z) = 2πi,8

and hence the identity (33) holds.For the other cases, the method of proof is similar and we shall no longer mention the details. For (14)with i = k ≤n −1, the identity to be verified is of the following type :kXi=1 Qkj=1[ai −bj][ai −cj −1]Qkj̸=i=1[ai −aj][ai −aj −1]−Qkj=1[ai −cj][ai −bj + 1]Qkj̸=i=1[ai −aj][ai −aj + 1]!=kXj=1(bj + cj −2aj),(35)or, using a similar transformation as before,kXi=1 Qkj=1(Ai −Bj)(Ai −q2Cj)AiQkj̸=i=1(Ai −Aj) Qkj=1(Ai −q2Aj)+Qkj=1(Ai −Cj)(Ai −q−2Bj)AiQkj̸=i=1(Ai −Aj) Qkj=1(Ai −q−2Aj)!=1 −Qkj=1(BjCj)Qkj=1 Aj2 .

(36)This identity is proven by taking the function f(z) = Qj(z −Bj)(z −q2Cj)/(z Qj(z −Aj)(z −q2Aj))and applying the same Residue theorem. Finally, the most complicated case is (14) with i = p > n. Theidentity to prove is (with s = p −n) :−sXi=1Qsj=1[ai −cj][ai −bj + 1]Qsj̸=i=1[ai −aj][ai −aj + 1]nYk=1[ai −dk −fk + 1][ai −dk + 1]+sXi=1Qsj=1[ai −bj][ai −cj −1]Qsj̸=i=1[ai −aj][ai −aj −1]nYk=1[ai −dk −fk][ai −dk]+nXk=1(fk)Qsj=1[dk −cj −1][dk −bj]Qsj=1[dk −aj −1][dk −aj]nYl̸=k=1[dk −dl −fl][dk −dl]=nXk=1fk +sXj=1(bj + cj −2aj).

(37)Herein, the fk are equal to θk,p−1 −θkp, and since the θ’s take only the values 0 and 1, the fk’s take onlythe values 0, ±1. To prove (37), one again has to use the same technique on a functionf(z) =Qsj=1(z −Bj)(z −q2Cj)z Qsj=1(z −Aj)(z −q2Aj)nYk=1(z −FkDk)(z −Dk) .However, it turns out that (37) is true as a general identity only when in the third summation the factor(fk) is replaced by [fk].

In the present case, this can be done without harm since for the values x = 0, ±1we have indeed that [x] = x. This completes the verification of the Cartan-Kac relations.For the e-Serre relations, the calculations are extremely lengthy, but when collecting terms with thesame Gel’fand-Zetlin basis vector and then taking apart the common factors, the remaining factor alwaysreduces to a simple finite expression which is easily verified to be zero.

These expressions always reduceto one of the following (trivial) identities :[a][b + 1] −[a + 1][b] = [a −b],(38)[a + 1] + [a −1] = [2][a],(39)1[a −1][a] +1[a][a + 1] =[2][a −1][a + 1]. (40)9

In fact, the last of these reduces to the second one, and in some sense the only identities needed to provethe e-Serre relations are (38) and (39), and combinations of them. Finally, the calculations for the f-Serrerelations are of a similar nature as those for the e-Serre relations.5CommentsWe have studied the class of essentially typical representations of the quantum superalgebra Uq[gl(n/m)]and connected the relations to be satisfied for these representations with certain q-identities.

At present,we do not know how to extend the present results to other finite-dimensional representations of Uq[gl(n/m)].In fact, also in the non-deformed case the problem of how to modify the classical analogs of (24–30) re-mains an open problem. There is some indication that for a typical representation the only modificationwould be to simply delete those terms for which the coefficient becomes undefined; however, this is stillunder investigation and we hope to report results in the future.

For atypical representations, the GZ basiswill presumably be no longer appropriate : if one still uses the same GZ-patterns in the case that [m] isatypical, it turns out that some |m⟩-vectors have a non-trivial projection both on the maximal submoduleand on the quotient module (of the module spanned by the GZ basis vectors with a modified action whenthe corresponding coefficient is undefined). This property was observed for atypical representations ofgl(2/2), and here again further investigations are under way.

Note that we also have not examined therepresentation theory of Uq[gl(n/m)] in the case of q being a root of unity.Acknowledgements. Two of us (T.D.P.

and N.I.S.) are grateful to Prof. Vanden Berghe for the possi-bility to work at the Department of Applied Mathematics and Computer Science, University of Ghent.Prof.

A. Salam is acknowledged for the kind hospitality offered to T.D.P. at the International Centerfor Theoretical Physics, Trieste.

We would like to thank Dr. Hans De Meyer (Ghent) for stimulatingdiscussions.This work was supported by the European Community, contract No. ERB-CIPA-CT92-2011 (Coop-eration in Science and Technology with Central and Eastern European Countries) and Grant Φ-215 ofthe Bulgarian Foundation for Scientific Research.References[1] Bracken A.J., Gould M.D., Zhang R.B.

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