Finite-Dimensional Representations of the Quantum Superalgebra $ U Q

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1 Introduction

This paper is devoted to the study of a class of finite-dimensional irreducible representations of the quantum 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 identities related to these representations. Quantum groups [5], finding their origin in the quantum inverse problem method [6] and in investigations related to the Yang-Baxter equation [15], have now become an important and widely used concept in 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 a Hopf algebra structure. The first example was given in [19, 30], and soon followed the generalization to any Kac-Moody Lie algebra with symmtrizable Cartan matrix [4, 12]. For the deformation of the enveloping algebra of a Lie superalgebra we mention the case of osp(1/2) [20, 21], later to be extended to Lie superalgebras 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) can

be deformed into irreducible representations of Uq[sl(n)] [13], and it was shown that one obtains all

finite-dimensional irreducible modules of Uq[sl(n)] in this way [27]. In [14], explicit expressions for the generators of Uq[sl(n)] acting on the “undeformed” Gel’fand-Zetlin basis were given. It is in the spirit of this work that our present paper should be seen. Here, we study a class of irreducible representations of the 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. Just as in the case of sl(n), the basis will remain undeformed, and the deformation will arise in the action of the quantum superalgebra generators on the basis vectors. In the literature, some results have already appeared for representations of quantum superalgebras of 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 module construction of Kac [17] was generalized to Uq[gl(n/m)] [35]. On the other hand, oscillator representations have 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 Lie superalgebra gl(n/m) and fix the notation. We also remind the reader of some representation theory of gl(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 basis introduced in [24] is written in explicit form. In the next section, we briefly recall the definition of the quantum superalgebra Uq[gl(n/m)]. Section 4 contains our main results. We present the actions of the

Uq[gl(n/m)] generators on the Gel’fand-Zetlin basis introduced, and we give some indications of how

2 the relations were proved in these representations. Some of the relations actually reduce to interesting q-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(n/m) and Gel’fand-Zetlin patterns

The Lie superalgebra G = gl(n/m) can be deﬁned [16, 28] through its natural matrix realization

A B gl(n/m)= {x = |A ∈ M ,B ∈ M , C ∈ M ,D ∈ M }, (1) CD n×n n×m m×n m×m where Mp×q is the space of all p × q complex matrices. The even subalgebra gl(n/m)0¯ has B = 0 and

C = 0; the odd subspace gl(n/m)1¯ has A = 0 and D = 0. The bracket is determined by

αβ [a,b]= ab − (−1) ba, ∀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 of

G = G0¯ ⊕ G1¯ is called homogeneous if it belongs to either G0¯ or else G1¯. We denote by gl(n/m)+1 the 0 B 0 0 space of matrices and by gl(n/m) the space of matrices . Then G = gl(n/m) 0 0 −1 C 0 Z Z has a -grading which is consistent with the 2-grading [28], namely G = G−1 ⊕ G0 ⊕ G+1 with G0¯ = G0 and G1¯ = 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 6= m) one can deﬁne the simple superalgebra sl(n/m) as the subalgebra consisting of elements with supertrace 0. However, the representation theory of gl(n/m) or sl(n/m) is essentially the same (the situation is similar as for the classical Lie algebras gl(n) and sl(n)), and hence we prefer to work with gl(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 and ∗ fi = 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). On H∗ there is a bilinear form deﬁned deduced from the supertrace on G, and explicitly given by [34]:

hǫi|ǫj i = δij , for 1 ≤ i, j ≤ n; hǫi|ǫpi =0, for 1 ≤ i ≤ n and n +1 ≤ p ≤ r; (3) hǫp|ǫqi = −δpq, for n +1 ≤ p, q ≤ r.

∗ where δij is the Kronecker-δ. The components of an element Λ ∈ H will be written as [m] = r ∗ [m1r,m2r,...,mrr] where Λ = i=1 mirǫi and mir are complex numbers. The elements of H are called the weights. The roots of glP(n/m) are the non-zero weights of the adjoint representation, and take the form ǫi − ǫj (i 6= j) in this basis; the positive roots are those with 1 ≤ i < j ≤ r, and of importance are the nm odd positive roots

βip = 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 6= n and an = mnr + mn+1,r. Hence, Λ with components [m] will

3 be called an integral dominant weight if mir − mi+1,r ∈ Z+ = {0, 1, 2,...} for all i 6= n (1 ≤ i ≤ r − 1). 0 For every integral dominant weight Λ ≡ [m] we denote by V (Λ) the simple G0 module with highest

weight Λ; this is simply the finite-dimensional gl(n) ⊕ gl(m) module with gl(n) labels {m1,r,...mnr} and 0 with gl(m) labels {mn+1,r,...,mrr}. The module V (Λ) can be extended to a G0 ⊕ G+1 module by the 0 requirement that G+1V (Λ) = 0. The induced G module V (Λ), first introduced by Kac [17] and usually referred to as the Kac-module, is defined by

G 0 ∼ 0 V (Λ) = IndG0⊕G+1 V (Λ) = U(G−1) ⊗ V (Λ), (5)

nm 0 where U(G−1) is the universal enveloping algebra of G−1. It follows that dim V (Λ) = 2 dim V (Λ). By deﬁnition, V (Λ) is a highest weight module; unfortunately, V (Λ) is not always a simple G module. It contains a unique maximal (proper) submodule M(Λ), and the quotient module

V (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 characterized by 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 typical

Kac module, resp. a typical simple module) if and only if hΛ+ ρ|βipi 6= 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. The importance 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 hΛ+ ρ|βipi = lir − lpr, and hence the conditions for typicality take a simple form. For typical modules or representations one can say that they are well understood, and a character formula was given by Kac [17]. A character formula for all atypical modules has not been proven so far, but there are several breakthroughs in this area : for singly atypical modules (for which the highest weight Λ is atypical with respect to one single odd root βip) a formula has been constructed [34]; for all atypical modules a formula has been conjectured [33]; for atypical Kac-modules the composition series has been conjectured [11] and partially shown to be correct [31]. On the other hand, the modules for which an explicit action of generators on basis vectors can be given, similar to the action of generators of gl(n) on basis vectors with Gel’fand-Zetlin labels, is only a subclass of the typical modules, namely the so-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 of subalgebras gl(n) ⊃ gl(n−1) ⊃···⊃ gl(1). In trying to construct a similar basis for the finite-dimensional modules of the Lie superalgebra gl(n/m) it was natural to consider the decomposition with respect to the

4 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 respect to this decomposition, it was necessary that at every step in this reduction the corresponding modules are completely reducible with respect to the submodule under consideration. A suﬃcient condition is that for every step in the above reduction the modules are typical, i.e. a typical gl(n/m) module V must decompose 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 be constructed with an action of the gl(m/n) generators. In terms of the above quantities lir, a module with highest 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 deduce it from relations (24–30) of the present paper by taking the limit q → 1 (in fact, the limit of our present relations 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 labelling of basis vectors of representations of the quantum algebra Uq[gl(n/m)] is exactly the same (note that also for the quantum algebra Uq[gl(n)], the ﬁnite-dimensional representations can be labelled by the same Gel’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 for gl(r): m1r ··· mn−1,r mnr mn+1,r ··· mr−1,r mrr m1,r−1 ··· mn−1,r−1 mn,r−1 mn+1,r−1 ··· mr−1,r−1  ......  m1,n+1 ··· mn−1,n+1 mn,n+1 mn+1,n+1  |m)=  (9) m ··· m − m  1n n 1,n nn  m1,n−1 ··· mn−1,n−1   . .  . ..   m  11   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 pattern to the right of the m × n rectangle mip (1 ≤ i ≤ n; n +1 ≤ p ≤ r) in (9) corresponds to a classical Gel’fand-Zetlin pattern for gl(m), and that the triangular pattern below this rectangle corresponds to a Gel’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 of the 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 shall then define an action of the quantum algebra generators on the Gel’fand-Zetlin basis vectors |m) introduced here. In other words, just as for the finite-dimensional gl(n) modules, one can use the same basis vectors and only the action is deformed.

3 The quantum superalgebra Uq[gl(n/m)]

The quantum superalgebra Uq ≡ Uq[gl(n/m)] is the free associative superalgebra over C with parameter C −1 q ∈ and generators kj , kj (j = 1, 2,...,r ≡ n + m) and ei, fi (i = 1, 2,...,r − 1) subject to the following relations (unless stated otherwise, the indices below run over all possible values) :

• The Cartan-Kac relations :

−1 −1 kikj = kj ki, kiki = ki ki =1; (11)

−1 (δij −δi,j+1)/2 −1 −(δij −δi,j+1)/2 kiej ki = q ej , kifj ki = q fj ; (12)

eifj − fjei = 0 if i 6= j; (13) 2 −2 2 −2 −1 eifi − fiei = (ki ki+1 − ki+1ki )/(q − q ) if i 6= n; (14) 2 2 −2 −2 −1 enfn + fnen = (knkn+1 − kn kn+1)/(q − q ); (15)

• The Serre relations for the ei (e-Serre relations) :

2 eiej = ej ei if |i − j| 6= 1; en =0; (16)

2 −1 2 ei ei+1 − (q + q )eiei+1ei + ei+1ei =0, for i ∈{1,...,n − 1}∪{n +1,...,n + m − 2}; (17) 2 −1 2 ei+1ei − (q + q )ei+1eiei+1 + eiei+1 =0, for i ∈{1,...,n − 2}∪{n,...,n + m − 2}; (18) −1 enen−1enen+1 +en−1enen+1en +enen+1enen−1 +en+1enen−1en −(q+q )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 the

requirement that the only odd generators are en and fn; the degree of a homogeneous element a of Uq

shall 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 , −1 −1 ∆(ei) = ei ⊗ kiki+1 + ki ki+1 ⊗ ei, if i 6= n, −1 −1 ∆(en) = en ⊗ knkn+1 + kn kn+1 ⊗ en, (21) −1 −1 ∆(fi) = fi ⊗ kiki+1 + ki ki+1 ⊗ fi, if i 6= n, −1 −1 ∆(fn) = fn ⊗ knkn+1 + kn kn+1 ⊗ fn; −1 S(kj ) = kj , −1 S(ei) = −qei, S(fi)= −q fi, if i 6= n, (22) S(en) = −en, S(fn)= −fn.

Remember that ∆ : Uq → Uq ⊗ Uq is a morphism of graded algebras, and that the multiplication in

Uq ⊗ Uq is given by (a ⊗ b)(c ⊗ d) = (−1)deg(b) deg(c)ac ⊗ bd. (23)

6 4 The Uq[gl(n/m)] representations

Let Λ ≡ [m] be an essentially typical highest weight, and denote by W (Λ) the vector space spanned by the basis vectors |m) of the form (9) satisfying the conditions (10). On this vector space, we shall deﬁne an 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 for n +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 weight

Λ can be deformed into an irreducible Uq[gl(n/m)] module W (Λ) with the same underlying vector space and with the action of the generators given by :

i i−1 mji− mj,i−1 /2 ki|m) = q j=1 j=1 |m), (1 ≤ i ≤ r), (24) kP k+1P k−1 1/2 i=1 [li,k+1 − ljk] i=1 [li,k−1 − ljk − 1] ek|m) = − k |m)jk, (1 ≤ k ≤ n − 1), (25) j=1 i=6 j=1[lik − ljk][lik − ljk − 1] ! X Q Q k k+1Q k−1 1/2 i=1 [li,k+1 − ljk + 1] i=1 [li,k−1 − ljk] fk|m) = − k |m)−jk, (1 ≤ k ≤ n − 1), (26) j=1 i=6 j=1[lik − ljk + 1][lik − ljk] ! X Q Q n Q n−1 1/2 [lk,n−1 − li,n+1] e |m) = θ (−1)i−1(−1)θ1n+...+θi−1,n k=1 |m) , (27) n in n [l − l ] in i=1 Qk=6 i=1 k,n+1 i,n+1 ! Xn i−1 θ1n+...+θi−Q1,n fn|m) = (1 − θin)(−1) (−1) [li,n+1 − ln+1,n+1] i=1 X n−1 1/2 k=1 [lk,n−1 − li,n+1] × |m)− , (28) n [l − l ] in Qk=6 i=1 k,n+1 i,n+1 ! n Q θ1p+...+θi−1,p+θi+1,p−1+...+θn,p−1 ep|m) = θip(−1) (1 − θi,p−1) i=1 X n 1/2 [li,p+1 − lkp][li,p+1 − lkp − 1] × |m)ip [l − l ][l − l − − 1] 6 i,p+1 k,p+1 i,p+1 k,p 1 k=Yi=1 p p−1 p+1 1/2 [lq,p−1 − lsp + 1] [lq,p+1 − lsp] + − q=n+1 q=n+1 p [l − l ][l − l + 1] s=n+1 Q q=6 s=n+1 qp spQqp sp ! nX [lkp − lsp][Qlkp − lsp + 1] × |m)sp, (n +1 ≤ p ≤ r − 1), (29) [lk,p+1 − lsp][lk,p−1 − lsp + 1] k=1 Yn θ1p+...+θi−1,p+θi+1,p−1+...+θn,p−1 fp|m) = θi,p−1(−1) (1 − θip) i=1 Xn [l − l ][l − l − 1] 1/2 × i,p+1 kp i,p+1 kp [l − l ][l − l − − 1] 6 i,p+1 k,p+1 i,p+1 k,p 1 k=Yi=1 p−1 p+1 q=n+1[li,p+1 − lq,p−1 − 1] q=n+1[li,p+1 − lq,p+1] × p |m)−ip Q q=n+1[li,p+1 − lqp −Q1][li,p+1 − lqp] Q

7 p p−1 p+1 1/2 q=n+1[lq,p−1 − lsp] q=n+1[lq,p+1 − lsp − 1] + − |m)− , p [l − l − 1][l − l ] sp s=n+1 Q q=6 s=n+1 qp Q sp qp sp ! X (n +1 Q≤ p ≤ r − 1). (30)

n n In the above expressions, k=6 i=1 or k=6 i=1 means that k takes all values from 1 to n with k 6= i. Ifa vector from the rhs of (24–30)P does notQ belong to the module under consideration, then the corresponding term is zero even if the coefficient in front is undefined; if an equal number of factors in numerator and denominator are simultaneously equal to zero, they should be cancelled out. The eqs. (25,26) are the same 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. Provided that all coefficients in (24–30) are well defined (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 f-Serre relations). The irreducibility 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 non-deformed matrix element vanishes. To show that (11), (12) and (13) are satisfied is a straighforward matter. The difficult Cartan-Kac relations to be verified are (14) and (15). We shall consider one case in more detail, namely (15). This relation, with the actions (24–30), is valid if and only if

n n−1[l − l ] [l − l ] k=1 k,n−1 i,n+1 i,n+1 n+1,n+1 n [l − l ] i=1 k=6 i=1 k,n+1 i,n+1 X Q n−1 Q = (lk,n+1 − lk,n−1)+ ln,n+1 − ln+1,n+1 . (31) " # Xk=1 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 identity between q-numbers to be proved reduces to

n n [a − b ] n k=1 i k = (a − b ) . (32) n [a − a ] k k i=1 k=6 i=1 i k " # X Q kX=1

Q 2ai 2bi Using the explicit deﬁnition of a q-number, and relabelling q = Ai and q = Bi, this becomes

n n k=1(Ai − Bk) B1B2 ··· Bn n =1 − . (33) A (A − A ) A1A2 ··· A i=1 i k=6 i=1 i k n X Q To prove this last identity, consider theQ complex function

n (z − B ) f(z)= k=1 k . (34) z n (z − A ) Q k=1 k

This function is holomorphic over C except in itsQ 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. Then n the Residue Theorem of complex analysis implies that C f(z)dz =2πi(Res(0)+ i=1 Res(Ai)). It is easy to see that Res(0) = lim f(z)z = (B ··· B )/(A ··· A ) and that Res(A ) = lim f(z)(z − A )= z→0 1 n 1 H n i P z→Ai i n n k=1(Ai − Bk)/(Ai k=6 i=1(Ai − Ak)). On the other hand, Q Q f(z)dz = −2πi Res(∞)= −2πi lim (−z)f(z)=2πi, z→∞ IC

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 :

k k k k [ai − bj ][ai − cj − 1] [ai − cj][ai − bj + 1] j=1 − j=1 = (b + c − 2a ) , (35) k k  j j j  i=1 Qj=6 i=1[ai − aj ][ai − aj − 1] Qj=6 i=1[ai − aj ][ai − aj + 1]! j=1 X X   or, using aQ similar transformation as before,Q

k k 2 k −2 (Ai − Bj)(Ai − q Cj ) (Ai − Cj )(Ai − q Bj ) j=1 + j=1 k k 2 k k −2 i=1 Ai jQ=6 i=1(Ai − Aj ) j=1(Ai − q Aj ) Ai jQ=6 i=1(Ai − Aj ) j=1(Ai − q Aj )! X k Qj=1(Bj Cj ) Q Q Q = 1 − 2 . (36) k Q j=1 Aj Q 2 2 This identity is proven by taking the function f(z) = j (z − Bj )(z − q Cj )/(z j (z − Aj )(z − q Aj )) and applying the same Residue theorem. Finally, the mostQ complicated case is (14)Q with i = p>n. The identity to prove is (with s = p − n):

s s n j=1[ai − cj ][ai − bj + 1] [ai − dk − fk + 1] − s [ai − aj ][ai − aj + 1] [ai − dk + 1] i=1 Qj=6 i=1 k=1 Xs s Yn Q j=1[ai − bj ][ai − cj − 1] [ai − dk − fk] + s [ai − aj ][ai − aj − 1] [ai − dk] i=1 Qj=6 i=1 k=1 Xn s Y n Q j=1[dk − cj − 1][dk − bj ] [dk − dl − fl] + (fk) s [dk − aj − 1][dk − aj ] [dk − dl] k=1 Qj=1 l=6 k=1 X Y n Q s = f + (b + c − 2a ) . (37)  k j j j  j=1 Xk=1 X   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 only the values 0, ±1. To prove (37), one again has to use the same technique on a function

s 2 n j=1(z − Bj )(z − q Cj ) (z − FkDk) f(z)= s 2 . z (z − Aj )(z − q Aj ) (z − Dk) Q j=1 k=1 Y However, it turns out that (37) is trueQ 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, ±1 we 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 the same Gel’fand-Zetlin basis vector and then taking apart the common factors, the remaining factor always reduces to a simple finite expression which is easily verified to be zero. These expressions always reduce to one of the following (trivial) identities :

[a][b + 1] − [a + 1][b] = [a − b], (38) [a + 1] + [a − 1] = [2][a], (39) 1 1 [2] + = . (40) [a − 1][a] [a][a + 1] [a − 1][a + 1]

9 In fact, the last of these reduces to the second one, and in some sense the only identities needed to prove the e-Serre relations are (38) and (39), and combinations of them. Finally, the calculations for the f-Serre relations are of a similar nature as those for the e-Serre relations.

5 Comments

We 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 modification would be to simply delete those terms for which the coefficient becomes undefined; however, this is still under investigation and we hope to report results in the future. For atypical representations, the GZ basis will presumably be no longer appropriate : if one still uses the same GZ-patterns in the case that [m] is atypical, it turns out that some |mi-vectors have a non-trivial projection both on the maximal submodule and on the quotient module (of the module spanned by the GZ basis vectors with a modified action when the corresponding coefficient is undefined). This property was observed for atypical representations of gl(2/2), and here again further investigations are under way. Note that we also have not examined the representation 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 Center for Theoretical Physics, Trieste. We would like to thank Dr. Hans De Meyer (Ghent) for stimulating discussions. 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 of the Bulgarian Foundation for Scientiﬁc Research.

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