The quantum determinant of the elliptic algebra A q,p ( gl ˆ N )
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The quantum determinantof the elliptic algebra A q,p ( b gl N ) L. Frappat , D. Issing and E. Ragoucy Univ. Grenoble Alpes, USMB, CNRS, F-74000 Annecy
Abstract
We introduce the quantum determinant for the elliptic quantum algebra A q,p ( b gl N )and prove that it generates the center of this algebra. We also show that it is group-likefor the quasi-Hopf structure, which allows us to define the elliptic quantum algebra A q,p ( b sl N ). [email protected] [email protected] [email protected] Address: BP 110 Annecy-le-Vieux, F-74941 Annecy Cedex, France.
Introduction
It is well known that certain algebraic structures are characterized by solutions of theYang–Baxter equation (YBE) with spectral parameter: for example, the rational and trigono-metric solutions lead to the Yangian and quantum affine algebras respectively. In this frame-work, called the FRT formalism [1], the generators of the algebra under consideration areencapsulated into the Lax matrix L ( z ), whose intertwining properties are coded in the R -matrix (RLL relations). Within this approach, Foda et al. [2,3] were able to define the ellipticquantum algebra A q,p ( b gl ) using the elliptic solution of the YBE ( R -matrix of the 8-vertexmodel) discovered by Baxter [4]. This solution exhibits entries expressed in terms of theJacobi theta function and depends on a deformation parameter q and an elliptic nome p .When properly normalized, Baxter’s elliptic R -matrix satisfies R ( q − ) = A where A is the antisymmetrizer on ( C ) ⊗ . This property implies a quantum determinant for-mula, such a quantum determinant lying in the center of the algebra. This allows to definethe elliptic algebra A q,p ( b sl ) by imposing further a restriction on the quantum determinant,namely A q,p ( b sl ) = A q,p ( b gl ) / h qdet L ( z ) − q c/ i (1.1)where c is the central charge of the algebra.The generalization to the b gl N case builds on the elliptic Z N -symmetric R -matrix derivedby Belavin [5]. This matrix reduces to Baxter’s elliptic R -matrix when N = 2. Along thesame lines as above, an elliptic quantum algebra A q,p ( b gl N ) was defined in [6] by using theelliptic Z N -symmetric R -matrix. However, an explicit formula for the quantum determinantis still missing in the case of A q,p ( b gl N ) when N >
2. The main result of this paper is thederivation of the quantum determinant formula for A q,p ( b gl N ) and the proof that it generatesthe center of the algebra. Hence formula (1.1) can be extended to generic N .Let us remark that this derivation has a slightly different final step, compared to the usualapproach used in the quantum affine or Yangian cases [1, 7, 8]. Indeed, the usual approachuses the fact that the antisymmetrizer on ( C N ) ⊗ N can be expressed as some product of R -matrices. In the elliptic case, this does not seem to be true when N >
2. It does not preventto apply the antisymmetrizer to an N -fold product of Lax matrices and to construct ananalog of the quantum determinant that is still central. However, it remains to compute thevalue of the quantum determinant in the fundamental evaluation representation. Withoutthe expression of the antisymmetrizer in term of R -matrices, it requires some more work withrespect to the usual case.The plan of the paper is as follows. In section 2, we recall the definition and basicproperties of the elliptic quantum algebra A q,p ( b gl N ). Section 3 is a reminder devoted to thequantum affine algebra U q ( b gl N ). Here, we note in particular that the limit p → R -matrix is a twisted version of the affine R -matrix in the principal gradation. Themain result of the paper, the formula for the quantum determinant of A q,p ( b gl N ), is stated insection 4. Proofs are gathered in section 5. A q,p ( b gl N ) A q,p ( b gl N ) Let us briefly review the construction of the elliptic quantum algebra A q,p ( b gl N ). Weconsider a free, associative, unital algebra generated by operators L i,j [ n ] (1 ≤ i, j ≤ N ,1 ∈ Z ), compactly represented by the formal series of the spectral parameter z ∈ C L i,j ( z ) = X n ∈ Z L i,j [ n ] z − n (2.1)and encapsulated into the so-called Lax matrix L ( z ) = N X i,j =1 L i,j ( z ) e i,j , where the matrices e i,j are the usual elementary matrices. We adjoin an invertible centralelement written as q c ( q is the deformation parameter and c the central charge).The A q,p ( b gl N ) algebra is defined by imposing a set of exchange relations (commonlyrefereed to as RLL or FRT relations [1]) on these generators: b R ( zw ) L ( z ) L ( w ) = L ( w ) L ( z ) b R ∗ ( zw ) . (2.2)The indices refer to the spaces in which the Lax operators and the matrix b R ( z ) operate, with L ( z ) = L ( z ) ⊗ I and L ( z ) = I ⊗ L ( z ). In addition to its spectral parameter dependence, the R -matrix b R ( z ) depends on two complex parameters q (deformation parameter) and p (ellipticnome). To lighten presentation, unless needed, we will not write explicitly the dependencein q and p . The second R -matrix appearing in (2.2) is related to the original one through b R ∗ ( z ) = b R ( z ) (cid:12)(cid:12) p → p ∗ = pq − c .The A q,p ( b gl N ) algebra is a quasi-Hopf algebra with coproduct∆ L ( z ; p ) = (cid:16) ⊗ L ( zq c (1) / ; p ) (cid:17) · (cid:16) L ( zq − c (2) / ; pq − c (2) ) ⊗ (cid:17) , (2.3)∆ c = c ⊗ ⊗ c ≡ c (1) + c (2) (2.4)where we have specified the p dependence in L ( z ), due to shifts occurring when we apply ∆(for more details see [9, 10]). One can easily check that ∆ is an algebra morphism.In order to define explicitly the matrix b R ( z ) appearing in here, we first introduce a slightlymodified matrix R ( z ) = P R b,da,c ( z ) e a,b ⊗ e c,d , whose non-vanishing entries obey a + c = b + d ,the addition of indices being understood modulo N . For 1 ≤ a, b, c, d ≤ N , one defines R b,da,c ( z ) = η ( z ) S ba,c ( z ) ω ( a + c − b − d ) / δ (mod N ) a + c,b + d , (2.5)where S ba,c ( z ) = z b − a ) N q c − b ) N p ( b − a )( c − b ) N Θ p N ( p N + c − a q z )Θ p N ( p N + c − b z )Θ p N ( p N + b − a q ) . (2.6)Remark that since ω = e iπ/N , the factor ω ( a + c − b − d ) / equals ±
1. This factor results from thegauge transformation done on the Belavin R -matrix that allows one to recover the original A q,p ( b gl ) matrix. Note that in (2.6) one can consider any index c ∈ Z because of the identity S ba,c + N ( z ) = S ba,c ( z ).The normalization coefficient η ( z ) is given by η ( z ) = z N κ N ( z ) ( p N , p N ) ∞ ( p, p ) ∞ Θ p ( q )Θ p ( pz )Θ p ( q z ) (2.7)2ith 1 κ N ( z ) = ( q N z − ; p, q N ) ∞ ( q z ; p, q N ) ∞ ( pz − ; p, q N ) ∞ ( pq N − z ; p, q N ) ∞ ( q N z ; p, q N ) ∞ ( q z − ; p, q N ) ∞ ( pz ; p, q N ) ∞ ( pq N − z − ; p, q N ) ∞ . (2.8)The matrix S is Z N -symmetric by construction, i.e. the coefficients satisfy S b + na + n,c + n ( z ) = S ba,c ( z ), n = 1 , ..., N , where again the addition of indices is modulo N .Here, Θ p ( z ) denotes the Jacobi theta function defined by ( p ∈ C such that | p | < p ( z ) = ( z ; p ) ∞ ( pz − ; p ) ∞ ( p ; p ) ∞ (2.9)and the infinite q -Pochhammer symbols are given by( z ; p , . . . , p m ) ∞ = Y n i ≥ (1 − zp n . . . p n m m ) . (2.10)It is easy to show that the Jacobi theta function enjoys the following properties:Θ a ( az ) = Θ a ( z − ) and Θ a ( a n z ) = ( − n z n a n ( n − / Θ a ( z ) , ∀ n ∈ Z , (2.11)Θ a ( az ) = Θ a ( az − ) . (2.12)The matrices b R ( z ) and R ( z ) differ only by a suitable normalization, which reads b R ( z ) = τ N ( q z − ) R ( z ) , (2.13)where τ N ( z ) = z N − Θ q N ( qz )Θ q N ( qz − ) . (2.14) Example: the A q,p ( b gl ) algebra Specifying N = 2, the R -matrix (2.6) reads explicitly as: R ( z ) = 1 κ ( z ) ( p ; p ) ∞ ( p ; p ) ∞ a ( z ) 0 0 d ( z )0 b ( z ) c ( z ) 00 c ( z ) b ( z ) 0 d ( z ) 0 0 a ( z ) (2.15)and the entries take the following form: a ( z ) = z − Θ p ( pz )Θ p ( pq )Θ p ( pq z ) ,b ( z ) = qz − Θ p ( z )Θ p ( pq )Θ p ( q z ) , c ( z ) = Θ p ( pz )Θ p ( q )Θ p ( q z ) ,d ( z ) = − p qz Θ p ( z )Θ p ( q )Θ p ( pq z ) . (2.16)The Z -symmetry of the R -matrix corresponds to a symmetry w.r.t. the diagonal and w.r.t.the anti-diagonal. It implies in particular that R ( z ) t t = R ( z ). This is no longer the casefor higher rank algebras. Remark 2.1
It has been shown [9, 11] that the elliptic quantum algebras are quasi-Hopfalgebras, obtained by a twisting procedure on the quantum affine algebra U q ( b gl N ). Theexplicit expression of the Drinfel’d twist and the proof that this twist indeed satisfies theshifted cocycle condition were given in [9], see also [12]. In the case of A q,p ( b gl ), the evaluated R -matrix (2.15) was recovered through this procedure. Note that for N >
2, the explicitexpression of the twist in the N -dimensional evaluation representation is, to our knowledge,still unknown. 3 .2 Properties of the R -matrix Let g and h be the matrices of order N defined by g ij = ω i δ ij and h ij = δ i +1 ,j for1 ≤ i, j ≤ N with ω = e iπ/N , the addition of indices being understood modulo N .We recall the following properties of the R -matrix R ( z ) [13, 14]: • Yang–Baxter equation (also holds for b R ( z )): R ( z z ) R ( z z ) R ( z z ) = R ( z z ) R ( z z ) R ( z z ) . (2.17) • Unitarity: R ( z ) R ( z − ) = I , (2.18) • Regularity ( P is the permutation matrix): R (1) = P , (2.19) • Crossing-symmetry: R ( z ) t R ( z − q − N ) t = I , (2.20) • Antisymmetry: R ( − z ) = ω ( g − ⊗ I ) R ( z ) ( g ⊗ I ) , (2.21) • Quasi-periodicity: b R ( − zp ) = ( g hg ⊗ I ) − b R ( z − ) − ( g hg ⊗ I ) , (2.22) • Invariance: ( h ⊗ h ) R ( z ) = R ( z ) ( h ⊗ h ) . (2.23) Remark 2.2
The crossing-symmetry and the unitarity properties of R allow to exchangethe inversion and the transposition when applied to the matrix R (or to the matrix b R ).It provides a crossing-unitarity relation (also valid for b R thanks to the q N -periodicity of thefunction τ N ): (cid:16) R ( x ) t (cid:17) − = (cid:16) R ( q N x ) − (cid:17) t . (2.24)Note also that the unitarity property for b R reads b R ( z ) b R ( z − ) = τ N ( q z ) τ N ( q z − ) ≡ U ( z ) , (2.25)where the function U ( z ) is defined as U ( z ) = q N − Θ q N ( q z ) Θ q N ( q z − )Θ q N ( z )Θ q N ( z − ) . (2.26) U q ( b gl N ) We recall in this section the different gradations that may be used in constructing the R -matrix defining the algebra U q ( b gl N ) in the FRT formalism. As will be shown below, the R -matrix obtained as the non-elliptic limit of (2.13) appears to be a twisted version of the R -matrix in the principal gradation. 4 .1 Homogeneous gradation. In the FRT formalism, the quantum affine algebra U q ( b gl N ) is described as an associativealgebra defined by generators and relations. The generators L ± i,j [ ∓ n ], where n ∈ Z ≥ , 1 ≤ i, j ≤ N and L + i,j [0] = L − j,i [0] = 0 for i > j , are coded in formal generating functions L ± i,j ( z ),themselves encapsulated into the Lax matrices L ± ( z ): L ± ( z ) = N X i,j =1 L ± i,j ( z ) e i,j and L ± i,j ( z ) = ∞ X n =0 L ± i,j [ ∓ n ] z ± n . (3.1)The relations are the well-known RLL relations R ( z ± w ± ) L ± ( z ) L ± ( w ) = L ± ( w ) L ± ( z ) R ( z ± w ± ) , (3.2) R ( z ± w ∓ ) L +1 ( z ) L − ( w ) = L − ( w ) L +1 ( z ) R ( z ∓ w ± ) , (3.3)where z ± = zq ± c/ , w ± = wq ± c/ , c is the central charge and the matrix R ( z ) is given by R ( z ) = ρ N ( z ) "X i e i,i ⊗ e i,i + q (1 − z )1 − q z X i = j e i,i ⊗ e j,j + (1 − q )1 − q z X i
1) and h i (0 ≤ i ≤ N ) denote the generators of U q ( b gl N ) in theSerre–Chevalley basis and let R be the universal R -matrix of U q ( b gl N ) (see e.g. [15]). The R -matrix (3.4) is obtained from R by calculating its image R ( z/w ) = ( π z ⊗ π w ) R in the N -dimensional evaluation representation π z such that (1 ≤ i ≤ N ) π z ( e i ) = e i,i +1 , π z ( f i ) = e i +1 ,i , π z ( h i ) = e i,i , (3.6) π z ( e ) = ze N, , π z ( f ) = z − e ,N , π z ( h ) = e N,N − e , . (3.7)This defines the so-called homogeneous gradation.The quantum affine algebra U q ( b gl N ) is endowed with the following coproduct structure:∆ (cid:0) L ± i,j ( z ) (cid:1) = N X k =1 L ± k,j ( zq ∓ c (2) / ) ⊗ L ± i,k ( zq ± c (1) / ) , (3.8)where c (1) = c ⊗ c (2) = 1 ⊗ c .The quantum determinant is given in the homogeneous gradation byqdet L ( z ) = X σ ∈ S N sgn( σ ) q ℓ ( σ ) L +1 ,σ (1) ( z ) L +2 ,σ (2) ( zq − ) . . . L + N,σ ( N ) ( zq − N ) , (3.9)where ℓ ( σ ) denotes the length of the permutation σ and sgn( σ ) = ( − ℓ ( σ ) .Finally, thanks to the RLL relations, the action of the finite Cartan generators on the Laxmatrices is given by (1 ≤ i, j, k ≤ N ) q h i L ± j,k ( w ) = L ± j,k ( w ) q h i + δ ik − δ ij . (3.10)5 .2 Principal gradation Another possible choice is the principal gradation. In that case, the evaluation map π z isgiven by π z ( e i ) = z /N e i,i +1 , π z ( f i ) = z − /N e i +1 ,i , π z ( h i ) = e i,i , (3.11) π z ( e ) = z /N e N, , π z ( f ) = z − /N e ,N , π z ( h ) = e N,N − e , . (3.12)The R -matrix in the principal gradation reads: R ( z ) = ρ N ( z ) "X i e i,i ⊗ e i,i + q (1 − z )1 − q z e i,i ⊗ e j,j + z (1 − q )1 − q z X i
2, this matrix differs from the previous one, essentially by some powers of q inthe diagonal terms. Obviously, it is still Z N -symmetric. It can be obtained from (3.13) by a(constant non factorized) diagonal twist: R ′ ( z ) = F R ( z ) F − (3.19)where F = N X i =1 e i,i ⊗ e i,i + X ≤ i = j ≤ N q α ij e i,i ⊗ e j,j (3.20)with, for i < j , α ij = + ( i − j ) /N and α ji = − α ij . We set by convention α ii = 0 for all i .Remark that for N = 2, α = 0, so that the twist is I ⊗ I .The algebra is still defined by Eqs. (3.2)–(3.1) where the Lax matrices L ± ( z ) are now replacedby L ′± ( z ).At the universal level, the twisted R -matrix is given by R F = F R F − (3.21)with F = q P ij α ij h i ⊗ h j . (3.22)Here h i ( i = 1 , . . . , N ) are the Cartan generators of the finite quantum algebra U q ( gl N )satisfying the following commutation relations ( j = 1 , . . . , N − h i , e j ] = ( δ ij − δ i,j +1 ) e j , [ h i , f j ] = − ( δ ij − δ i,j +1 ) f j , [ e j , f j ] = q h j − h j +1 − q h j +1 − h j q − q − . (3.23)The universal twist (3.22) satisfies the cocycle condition F (∆ ⊗ id) F = F (id ⊗ ∆) F ,ensuring that the universal R -matrix R F satisfies the Yang–Baxter equation while the R -matrix R does.The relation between the corresponding Lax matrices L ± and L ′± can be expressed as L ′± ( z ) = ( π z ⊗ id) F L ± ( z ) ( π z ⊗ id) F − . (3.24)In the evaluation representation π z , one gets( π z ⊗ id) F = ( π z ⊗ id) F − = N X i =1 q P Nj =1 α ij h j e i,i . (3.25)The twist being diagonal and depending only on the finite Cartan generators, the equation(3.10) also holds for the Lax matrices L ′± ( z ).The coproduct of the twisted algebra is given by ∆ F = F ∆ F − . A direct calculation showsthat ∆ F (cid:0) L ′± i,j ( z ) (cid:1) = N X k =1 L ′± k,j ( zq ∓ c (2) / ) ⊗ L ′± i,k ( zq ± c (1) / ) (3.26)from which it follows that the twisted algebra gets the same coproduct structure as theoriginal algebra.Applying the twist to the expression (3.17), and using the correspondence (3.24), we get an7xpression for the quantum determinant in this new presentation. It is again expressed as asum over permutations:qdet L ( z ) = X σ ∈ S N sgn( σ ) q n σ L ′ +1 ,σ (1) ( z ) L ′ +2 ,σ (2) ( zq − ) . . . L ′ + N,σ ( N ) ( zq − N ) , (3.27)where n σ = ℓ ( σ ) + N P Ni =1 i ( σ ( i ) − i ) + P ≤ i Theorem 4.1 Let A ( N ) N be the antisymmetrizer of N spaces C N . For generic values of theparameters p , q and of the central charge c , one has the following identity L ( z ) . . . L N ( zq − N ) A ( N ) N = qdet L ( z ) A ( N ) N , (4.1) where qdet L ( z ) , called the quantum determinant, is a scalar function that lies in the centerof the A q,p ( b gl N ) algebra. It can be rewritten as qdet L ( z ) = tr ...N (cid:16) L ( z ) . . . L N ( zq − N ) A ( N ) N (cid:17) (4.2)= X σ ∈ S N sgn( σ ) L ,σ (1) ( z ) L ,σ (2) ( zq ) . . . L N,σ ( N ) ( zq − N ) , (4.3) where S N is the set of permutations of N objects.Conversely, for generic values of the parameters p , q and of the central charge c , thequantum determinant generates the center of the A q,p ( b gl N ) algebra. The next section is devoted to the proofs of this theorem. By generic values of the parameters p , q and of the central charge c , we mean that there is no functional relation among them,such as the ones used in [6, 16] to define deformations of W N algebras, and that they do notobey any algebraic relation, such as c = − N or q being a root of unity, where it is knownthat the center is extended, see [17] for the former case and [18, 19] for the latter.Remark that for N = 2, it was already proven in [3] that the quantum determinant iscentral. To the best of our knowledge, the case N > p → 0, we recover the formula (3.28), as expected. Corollary 4.2 The quantum determinant is group-like: ∆qdet L ( z ) = qdet L ( zq − c (2) / ; pq − c (2) ) ⊗ qdet L ( zq c (1) / ; p ) , (4.4) where we have specified the p -dependence, as in (2.3) . roof: We apply the coproduct (2.3) to the expression (4.1). For brevity and for this calcu-lation, we note L ( z ) ≡ L ( z ; p ) and L ∗ ( z ) ≡ L ( z ; pq − c (2) ). We get∆qdet L ( z ) A ( N ) N = ∆ L ( z ) ∆ L ( zq − ) . . . ∆ L N ( zq − N ) A ( N ) N = (cid:16) ⊗ L ( zq c (1) / ) (cid:17)(cid:16) L ∗ ( zq − c (2) / ) ⊗ (cid:17) . . . (cid:16) ⊗ L N ( zq − N q c (1) / ) (cid:17)(cid:16) L ∗ N ( zq − N q − c (2) / ) ⊗ (cid:17) A ( N ) N = (cid:16) ⊗ L ( zq − c (1) / ) . . . L N ( zq − N q − c (1) / ) (cid:17)(cid:16) L ∗ ( zq c (2) / ) . . . L ∗ N ( zq − N q c (2) / ) ⊗ (cid:17) A ( N ) N = (cid:16) ⊗ L ( zq − c (1) / ) L ( zq − q − c (1) / ) . . . L N ( zq − N q − c (1) / ) A ( N ) N (cid:17) (cid:16) qdet L ∗ ( zq c (2) / ) ⊗ (cid:17) = (cid:16) ⊗ qdet L ( zq − c (1) / ) A ( N ) N (cid:17) (cid:16) qdet L ∗ ( zq c (2) / ) ⊗ (cid:17) = qdet L ∗ ( zq − c (2) / ) ⊗ qdet L ( zq c (1) / ) A ( N ) N where in the last step we have used that the quantum determinant is central.Theorem 4.1 and corollary 4.2 allow us to introduce the elliptic quantum algebra associ-ated to b sl N : Definition 4.3 The elliptic quantum algebra A q,p ( b sl N ) is the quasi-Hopf algebra defined bythe coset A q,p ( b sl N ) = A q,p ( b gl N ) / h qdet L ( z ) − q c/ i . (4.5) Proof: Since the quantum determinant and c are central, the coset defines an algebra. More-over, qdet L ( z ) and q c/ are both group-like, so that the coset is a quasi-Hopf algebra. The proof of the main theorem relies on different lemmas and properties. Lemma 5.1 Let A ( N )2 be the antisymmetrizer on ( C N ) . Then for generic values of q , ker b R ( q ) = im A ( N )2 . Proof: It has been shown in [14] that b R ( q )(1 − P ) = 0, which implies ker b R ( q ) ⊃ im A ( N )2 .Then, it remains to show that these two spaces have same dimension. Since the entries ofthe R -matrix of A q,p ( b gl N ) are products of (fractional) power of p and analytical functions of p , it is sufficient to consider the non-elliptic limit of the matrix, i.e. p → 0, leading to the R -matrix (3.18) of U q ( b gl N ).Denoting λ ′ = ρ N ( q ) − λ and Q = q q , one immediately gets thatdet( R ′ ( q ) − λ Id) ρ N ( q ) N = (1 − λ ′ ) N Y ≤ i In the A q,p ( b gl N ) algebra, the following identity holds L i,j ( z ) L k,l ( zq ) − L i,l ( z ) L k,j ( zq ) = L k,l ( z ) L i,j ( zq ) − L k,j ( z ) L i,l ( zq ) ∀ i, j, k, l = 1 , . . . , N. (5.2) In particular, we have L i,j ( z ) L i,l ( zq ) = L i,l ( z ) L i,j ( zq ) ∀ i, j, l = 1 , . . . , N. (5.3) Proof: We consider the RLL relations (2.2) for w = z/q and project them onto an arbitraryelement e i,j ⊗ e k,l . This leads to the following equation, valid for all i, j, k, l = 1 , . . . , N : N X n,m =1 b R n,mi,k ( q ) L n,j ( z ) L m,l ( zq ) = N X n,m =1 b R ∗ j,lm,n ( q ) L k,n ( zq ) L i,m ( z ) , (5.4)We will refer to this equation as X j,li,k ( z ).Note that the lemma 5.1 implies the relations b R j,li,k ( q ) = b R l,ji,k ( q ) . (5.5)Hence, looking at the difference X j,li,k ( z ) − X l,ji,k ( z ), the R.H.S. is equal to N X n,m =1 (cid:16) b R ∗ j,lm,n ( q ) − b R ∗ l,jm,n ( q ) (cid:17) L k,n ( zq ) L i,m ( z ) = 0 (5.6)In other words, X j,li,k ( z ) − X l,ji,k ( z ) does not depend on p ∗ . Finally, the L.H.S. gives us N X n,m =1 b R n,mi,k ( q ) (cid:18) L n,j ( z ) L m,l ( zq ) − L n,l ( z ) L m,j ( zq ) (cid:19) = 0 . (5.7)Note that the indices j and l do not play any role in these relations, so if we can solve (5.7)for one pair j, l , we can do it for any. We thus consider the equations for fixed indices j and l , and omit them to ease the notation.Denoting T n,m ( z ) ≡ T j,ln,m ( z ) = L n,j ( z ) L m,l ( zq ) − L n,l ( z ) L m,j ( zq ) + ( n ↔ m ) , (5.8)one gets for fixed i, j, k, l , using again the property (5.5), N X n,m =1 b R n,mi,k ( q ) T n,m ( z ) = 0 . (5.9)Since T ( z ) is in the symmetric part of C N ⊗ C N , lemma 5.1 implies that the only solution ofthe linear system (5.9) is T n,m ( z ) = 0, which is relation (5.2).10 .2 Explicit expression of the quantum determinant The antisymmetrizer A ( N ) N in ( C N ) ⊗ N is a rank 1 projector, the eigenvector correspondingto the eigenvalue 1 being given by w = X σ ∈ S N sgn( σ ) e σ (1) ⊗ · · · ⊗ e σ ( N ) , (5.10)where { e i } denotes the standard vector basis of C N . A ( N ) N projects any given vector v on w : A N v = h w , v i w ∀ v ∈ ( C N ) ⊗ N , (5.11)where h w , v i = P σ ∈ S N sgn( σ ) v σ (1) ...σ ( N ) .Due to equality (5.11), to get an expression for the quantum determinant, it is enough tocompute L ( z ) . . . L N ( zq − N ) w = N X i ,...,i N =1 X σ ∈ S N sgn( σ ) L i ,σ (1) ( z ) . . . L i N ,σ ( N ) ( q − N z )( e i ⊗ · · · ⊗ e i N ) . (5.12)We first prove that all the indices i , . . . , i N in (5.12) must be different. For such a purpose,we prove that terms with identical indices vanish. The proof is done by recursion on the’distance’ between two identical indices.Consider the terms with i k = i k +1 . Without loss of generality, we can check what happensfor k = N − 1, the reasoning naturally translates to all other possible pairs of adjacent indices.Focusing on the coefficient of e i ⊗ · · · ⊗ e i N − ⊗ e i N ⊗ e i N only, we write (all indices arbitrary,but fixed) X σ ∈ S N sgn( σ ) L i ,σ (1) ( z ) . . . L i N ,σ ( N − ( q − N z ) L i N ,σ ( N ) ( q − N z ) == X σ ′ ∈ S N sgn( σ ′ ◦ s N,N − ) L i ,σ ′ (1) ( z ) . . . L i N ,σ ′ ( N ) ( q − N z ) L i N ,σ ′ ( N − ( q − N z )= 12 X σ ′ ∈ S N sgn( σ ′ ) L i ,σ ′ (1) ( z ) . . . L i N − ,σ ′ ( N − ( q − N z ) ×× (cid:16) L i N ,σ ′ ( N − ( q − N z ) L i N ,σ ′ ( N ) ( q − N z ) − L i N ,σ ′ ( N ) ( q − N z ) L i N ,σ ′ ( N − ( q − N z ) (cid:17) = 0 (5.13)where the last equality is done by virtue of (5.3).Suppose now that the terms where i k = i k + n with 1 ≤ n ≤ m have zero contribution and11onsider the term where i k = i k + m +1 : X σ ∈ S N sgn( σ ) L i ,σ (1) ( z ) . . . L i k ,σ ( k ) ( q − k z ) . . . L i k ,σ ( k + m +1) ( q − m − k z ) . . . L i N ,σ ( N ) ( q − N z )= 12 X σ ′ ∈ S N sgn( σ ′ ) L i ,σ ′ (1) ( z ) . . . L i k − ,σ ′ ( k − ( q − k z ) × (cid:16) L i k ,σ ′ ( k ) ( q − k z ) L i k +1 ,σ ′ ( k +1) ( q − k z ) − L i k ,σ ′ ( k +1) ( q − k z ) L i k +1 ,σ ′ ( k ) ( q − k z ) (cid:17) × L i k +2 ,σ ′ ( k +2) ( q − k z ) . . . L i k ,σ ′ ( k + m +1) ( q − m − k z ) . . . L i N ,σ ′ ( N ) ( q − N z ) (5.14)= − X σ ′ ∈ S N sgn( σ ′ ) L i ,σ ′ (1) ( z ) . . . L i k − ,σ ′ ( k − ( q − k z ) × (cid:16) L i k +1 ,σ ′ ( k ) ( q − k z ) L i k ,σ ′ ( k +1) ( q − k z ) − L i k +1 ,σ ′ ( k +1) ( q − k z ) L i k ,σ ′ ( k ) ( q − k z ) (cid:17) × L i k +2 ,σ ′ ( k +2) ( q − k z ) . . . L i k ,σ ′ ( k + m +1) ( q − m − k z ) . . . L i N ,σ ′ ( N ) ( q − N z ) (5.15)= − X σ ∈ S N sgn( σ ) L i ′ ,σ (1) ( z ) . . . L i ′ k ,σ ( k ) ( q − k z ) . . . L i ′ k ,σ ( k + m ) ( q − m − k z ) . . . L i ′ N ,σ ( N ) ( q − N z ) . (5.16)To get (5.14), we have used the same trick as in the calculation of (5.13). To go from (5.14)to (5.15) we have used the relation (5.2). In the last equality, we have introduced the indices i ′ ℓ = i ℓ for ℓ / ∈ { k, k + 1 } and i ′ k = i k +1 , i ′ k +1 = i k . This last expression vanishes due to therecursion hypothesis.Since all indices i r are different, we can replace the sum on i , . . . , i N by a sum overpermutations µ ∈ S N . We pick one such permutation and examine the coefficient of e µ (1) ⊗· · · ⊗ e µ ( N ) : χ µ := X σ ∈ S N sgn( σ ) L µ (1) ,σ (1) . . . L µ ( N ) ,σ ( N ) (5.17)= 12 X σ ∈ S N sgn( σ ) L µ (1) ,σ (1) . . . L µ ( k − ,σ ( k − × (cid:8) L µ ( k ) ,σ ( k ) L µ ( k +1) ,σ ( k +1) − L µ ( k ) ,σ ( k +1) L µ ( k +1) ,σ ( k ) (cid:9) L µ ( k +2) ,σ ( k +2) . . . L µ ( N ) ,σ ( N ) . (5.18)But we can also look at a different permutation µ ′ = µ ◦ s k . In this case, we find that χ ( µ ◦ s k ) = 12 X σ ∈ S N sgn( σ ) L µ (1) ,σ (1) . . . L µ ( k − ,σ ( k − × (cid:8) L µ ( k +1) ,σ ( k ) L µ ( k ) ,σ ( k +1) − L µ ( k +1) ,σ ( k +1) L µ ( k ) ,σ ( k ) (cid:9) L µ ( k +2) ,σ ( k +2) . . . L µ ( N ) ,σ ( N ) . (5.19)Once more, condition (5.2) shows that χ ( µ ◦ s k ) = − χ µ . This allows us to conclude that infact, for any σ, µ ∈ S N , we have χ µ = sgn( σ ) χ ( µ ◦ σ ) . In particular, χ µ = sgn( µ ) χ id , and wefinally arrive at the following result: L ( z ) . . . L N ( zq − N ) w = 1 N ! X µ ∈ S N χ µ e µ (1) ⊗ · · · ⊗ e µ ( N ) (5.20)= 1 N ! X µ ∈ S N sgn( µ ) χ id e µ (1) ⊗ · · · ⊗ e µ ( N ) = χ id w . (5.21)12rom this, we directly infer that the quantum determinant is χ id , which proves the equality(4.3).Remark that in this way we have proved that L ( z ) . . . L N ( zq − N ) A ( N )1 ...N = M ( z ) A ( N )1 ...N , (5.22)where M ( z ) is scalar in the spaces 1,..., N and given by (4.3). It remains to prove that M ( z )is central in A q,p ( b gl N ): it is done in section 5.4. L ( u ) in the fundamental representation Lemma 5.3 The R -matrix (2.6) for the A q,p ( b gl N ) algebra obeys the following relation b R ( z ) . . . b R N (cid:0) zq − N (cid:1) A ( N )1 ...N = A ( N )1 ...N . (5.23) Proof: We apply the evaluation maps π j : L j ( z ) → b R j ( z ) , j = 1 , ..., N to the equality (5.22): b R ( z ) . . . b R N (cid:0) zq − N (cid:1) A ( N )1 ...N = π (cid:0) qdet L ( z ) (cid:1) A ( N )1 ...N = M ( z ) A ( N )1 ...N , (5.24)where π = π ⊗ π ⊗ ... ⊗ π N and M ( z ) is a matrix M ( z ) (yet to be determined) actingon the space 0 only. One can show from the evaluation of the relation (4.3) that M ( z ) is adiagonal matrix: M ( z ) = N − Y j =0 η (cid:18) zq j (cid:19) N X k =1 q k − N − X σ ∈ S N sgn( σ ) b S σ (1)1 ,k ( z ) . . . b S σ ( N ) N,k + P N − i =1 ( i − σ ( i )) (cid:18) zq N − (cid:19) e k,k ≡ N X k =1 m k ( z ) e k,k , (5.25)where we set, for any indices a, b, c , b S ba,c ( z ) := Θ p N ( p N + c − a q z )Θ p N ( p N + c − b z )Θ p N ( p N + b − a q ) . (5.26)Using the expression of η ( z ), we get m k ( z ) = (cid:18) − ( p N ; p N ) ∞ ( p ; p ) ∞ (cid:19) N q k − N Θ p ( q ) N Θ p ( z )Θ p ( q z ) X σ ∈ S N sgn( σ ) N Y ℓ =1 b S σ ( ℓ ) ℓ,k + P ℓ − i =1 ( i − σ ( i )) ( zq ℓ − ) . (5.27)Now, from the invariance property (2.23) of the R -matrix, it is easy to show that h h · · · h N π (cid:0) qdet L ( z ) (cid:1) A ( N )1 ...N = π (cid:0) qdet L ( z ) (cid:1) h h · · · h N A ( N )1 ...N = π (cid:0) qdet L ( z ) (cid:1) A ( N )1 ...N h h · · · h N . (5.28)Due to the expression (5.24), it implies [ h , M ( z )] = 0 which can be recasted as m k ( z ) = m k +1 ( z ), that is to say M ( z ) = m ( z ) I . 13sing property (2.11), it is easy to show that m ( z ; qp N/ , p ) = m ( z ; q, p ), where we haveindicated explicitly the dependence in the parameters q and p . In other words, since | p | < m ( z ; q, p ) = m ( z ; qp ℓN/ , p ) = lim ℓ →∞ m ( z ; qp ℓN/ , p ) = lim q ′ → m ( z ; q ′ , p ) , ∀ p, q. (5.29)This shows that m ( z ; q, p ) does not depend on q . To compute it, we take the limit q → 1. Toshow that this limit is well-defined, we computed explicitly relation (5.23) in the limit p → q and N (it corresponds to the non-elliptic presentation of U q ( b gl N ), seesection 3.3). We got m ( z ; q, p ) | p =0 = 1. Since p , q and c are generic, it shows that the limit(5.29) exists at least in a neighborhood of p = 0. Due to the term Θ p ( q ) N in (5.27), whichvanishes in the limit q → 1, one sees that only the term σ = id contributes to m ( z ; q, p ).Then, a direct calculation shows that m ( z ; 1 , p ) = 1 for generic values of p and N .We checked relation (5.23) for N = 2 , p and q . We wish to show that, for z, w ∈ C [qdet L ( z ) , L ( w )] = 0 . (5.30)This will be achieved by commuting L ( w ) through the expression (4.2) for the quantumdeterminant:qdet L ( z ) L ( w ) = tr ...N h L ( z ) . . . L N ( zq − N ) L ( w ) A ( N )1 ...N i = tr ...N h L ( z ) . . . L N − ( zq − N ) b R − N (cid:16) zw q − N (cid:17) L ( w ) L N ( zq − N ) b R ∗ N (cid:16) zw q − N (cid:17) A ( N )1 ...N i = tr ...N h b R − N (cid:16) zw q − N (cid:17) . . . b R − (cid:16) zw (cid:17) L ( w ) L ( z ) . . . L N ( zq − N ) × b R ∗ (cid:16) zw (cid:17) . . . b R ∗ N (cid:16) zw q − N (cid:17) A ( N )1 ...N i = tr ...N h b R − N (cid:16) zw q − N (cid:17) . . . b R − (cid:16) zw (cid:17) L ( w ) L ( z ) . . . L N ( zq − N ) A ( N )1 ...N i , = tr ...N h b R − N (cid:16) zw q − N (cid:17) . . . b R − (cid:16) zw (cid:17) L ( w ) qdet L ( z ) A ( N )1 ...N i , (5.31)where we used the RLL relations (2.2) and the fact that generators acting in different sub-spaces commute. The last equalities are due to lemma 5.3 and definition 4.1.Next, using the fact that the quantum determinant is a scalar in the spaces 0 , , . . . , N , weget qdet L ( z ) L ( w ) = tr ...N h b R − N (cid:16) zw q − N (cid:17) . . . b R − (cid:16) zw (cid:17) A ( N )1 ...N L ( w ) i qdet L ( z )= L ( w )tr ...N h A ( N )1 ...N i qdet L ( z )= L ( w ) qdet L ( z ) , (5.32)where we used that L ( w ) and the antisymmetrizer commute as they live in different spaces,applied the inverse of (5.23), and finally traced over the antisymmetrizer. As a consequence,the quantum determinant lies in the center of the algebra A q,p ( b gl N ) as desired.14 .5 Center of the algebra A q,p ( b gl N ) It is known that in U q ( b gl N ) and for generic values of q and c , the quantum determinant,as defined in (3.9), generates the center of this quantum algebra [1]. Since, being a twist of it, A q,p ( b gl N ) is isomorphic to U q ( b gl N ) as an algebra [9] and for generic values of p , q and c (in thesense explained in section 4), (3.9) also describes the full center of the algebra A q,p ( b gl N ). Thesame is true for the other two expressions (3.17) and (3.28), that are just the same quantumdeterminant in different presentations.Moreover, we have shown that expression (4.3) also lies in the center of A q,p ( b gl N ), andthat its limit for p → U q ( b gl N ) in the non-elliptic presentation, expression (3.28). 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