The elliptic GL(n) dynamical quantum group as an h-Hopf algebroid

Using the language of h-Hopf algebroids which was introduced by Etingof and Varchenko, we construct a dynamical quantum group, F_ell(GL(n)), from the elliptic solution of the quantum dynamical Yang-Baxter equation with spectral parameter associated to the Lie algebra sl_n. We apply the generalized FRST construction and obtain an h-bialgebroid F_ell(M(n)). Natural analogs of the exterior algebra and their matrix elements, elliptic minors, are defined and studied. We show how to use the cobraiding to prove that the elliptic determinant is central. Localizing at this determinant and constructing an antipode we obtain the h-Hopf algebroid F_ell(GL(n)).


Introduction
The quantum dynamical Yang-Baxter (QDYB) equation was introduced by Gervais and Neveu [GN84]. It was realized by Felder [F95] that this equation is equivalent to the Star-Triangle relation in statistical mechanics. It is a generalization of the quantum Yang-Baxter equation, involving an extra, so called dynamical, parameter. In [F95] an interesting elliptic solution to the QDYB equation with spectral parameter was given, adapted from the A (1) n solution to the Star-Triangle relation constructed in [JKMO88]. Felder also defined a tensor category, which he suggested should be thought of as an elliptic analog of the category of representations of quantum groups. This category was further studied in [FV96] in the sl 2 case.
In [FV97], the authors considered objects in Felder's category which were proposed as analogs of exterior and symmetric powers of the vector representation of gl n . To each object in the tensor category they associate an algebra of vector-valued difference operators and prove that a certain operator, constructed from the analog of the top exterior power, commutes with all other difference operators. This is also proved in [TV01] (Appendix B) in more detail and in [ZSY03] using a different approach.
An algebraic framework for studying dynamical R-matrices without spectral parameter was introduced in [EV98]. There the authors defined the notion of h-bialgebroids and h-Hopf algebroids, a special case of the Hopf algebroids defined by Lu [L96]. They also show, using a generalized version of the FRST construction, how to associate to every solution R of the non-spectral quantum dynamical Yang-Baxter equation an h-bialgebroid. Under some extra condition they get an h-Hopf algebroid by adjoining formally the matrix elements of the inverse L-matrix. This correspondence gives a tensor equivalence between the category of representations of the R-matrix and the category of so called dynamical representations of the h-bialgebroid.
In this paper we define an h-Hopf algebroid associated to the elliptic R-matrix from [F95] with both dynamical and spectral parameter for g = sl n . This generalizes the spectral elliptic dynamical GL(2) quantum group from [KNR04] and the non-spectral trigonometric dynamical GL(n) quantum group from [KN06]. As in [KNR04], this is done by first using the the generalized FRST construction, modified to also include spectral parameters. In addition to the usual RLL-relation, residual relations must be added "by hand" to be able to prove that different expressions for the determinant are equal.
Instead of adjoining formally all the matrix elements of the inverse L-matrix, we adjoin only the inverse of the determinant, as in [KNR04]. Then we express the antipode using this inverse. The main problem is to find the correct formula for the determinant, to prove that it is central, and to provide row and column expansion formulas for the determinant in the setting of h-bialgebroids.
The plan of this paper is as follows. After introducing some notation in Section 2.1, we recall the definition of the elliptic R-matrix in Section 2.2. In Section 3 we review the definition of h-bialgebroids and the generalized FRST construction with special emphasize on how to treat residual relations for a general R-matrix. We write down the relations explicitly in Section 4 for the algebra F ell (M (n)) obtained from the elliptic R-matrix. In particular we show that only one family of residual identities are needed.
Left and right analogs of the exterior algebra over C n is defined in Section 5 in a similar way as in [KN06]. They are certain comodule algebras over F ell (M (n)) and arise naturally from a single relation analogous to v ∧ v = 0. The matrix elements of these corepresentations are generalized minors depending on a spectral parameter. Their properties are studied in Section 6. In particular we show that the left and right versions of the minors in fact coincide. In Section 6.3 we prove Laplace expansion formulas for these elliptic quantum minors.
In Section 7 we show that the h-bialgebroid F ell (M (n)) can be equipped with a cobraiding, in the sense of [R04]. We use this and the the ideas as in [FV97] and [TV01] to prove that the determinant is central for all values of the spectral parameters. This implies that the determinant is central in the operator algebra as shown in [FV97].
Finally, in Section 7.4 we define F ell (GL(n)) to be the localization of F ell (M (n)) at the determinant and show that it has an antipode giving it the structure of an h-Hopf algebroid.

Notation
Let p, q ∈ R, 0 < p, q < 1. We assume p, q are generic in the sense that if p a q b = 1 for some a, b ∈ Z, then a = b = 0.
Recall also the Jacobi triple product identity, which can be written (1 − p j ). (2.4) It will sometimes be convenient to use the auxiliary function E given by E : C → C, E(s) = q s θ(q −2s ). (2.5) as meromorphic functions on h * × C × with values in End h (V ⊗ V ).
In the example we study, h is the Cartan subalgebra of sl(n). Thus h is the abelian Lie algebra of all traceless diagonal complex n × n matrices. Let V be the h-module C n with standard basis e 1 , . . . , e n . Define ω(i) ∈ h * (i = 1, . . . , n) by (2.10) For the readers convenience, we give the explicit relationship between the R-matrix (2.8) and Felders R-matrix as written in [FV97] which we denote by R 1 . Thus R 1 : where h 1 is the Cartan subalgebra of gl(n), is defined as in (2.8) with α, β replaced by α 1 , β 1 : C 2 → C, (2.12) Here τ, γ ∈ C with Im τ > 0 and θ 1 is the first Jacobi theta function As proved in [F95], R 1 satisfies the following version of the QDYBE: and the unitarity condition (2.14) We can identify h * ≃ h * 1 /C tr where tr ∈ h * 1 is the trace. Since R 1 has the form (2.8), it is constant, as a function of λ ∈ h * 1 , on the cosets modulo C tr. So R 1 induces a map h * × C → End(V ⊗ V ), which we also denote by R 1 , still satisfying (2.13),(2.14).

Definitions
We recall the some definitions from [EV98]. Let h * be a finite-dimensional complex vector space (for example the dual space of an abelian Lie algebra) and M h * be the field of meromorphic functions on h * .
Definition 3.1. An h-algebra is a complex associative algebra A with 1 which is bigraded over h * , A = α,β∈h * A αβ , and equipped with two algebra embeddings µ l , µ r : M h * → A, called the left and right moment maps, such that A morphism of h-algebras is an algebra homomorphism preserving the bigrading and the moment maps.
The matrix tensor product Example 3.2. Let D h be the algebra of operators on M h * of the form i f i T α i with f i ∈ M h * and α i ∈ h * . It is an h-algebra with bigrading f T −α ∈ (D h ) αα and both moment maps equal to the natural embedding.
For any h-algebra A, there are canonical isomorphisms Definition 3.3. An h-bialgebroid is an h-algebra A equipped with two h-algebra morphisms, the comultiplication ∆ : A → A ⊗A and the counit ε :

The generalized FRST-construction
In [EV98] the authors gave a generalized FRST-construction which attaches an hbialgebroid to each solution of the quantum dynamical Yang-Baxter equation without spectral parameter. It was described in [KNR04] one way of extending this to the case when a spectral parameter is also present. However, when specifying the R-matrix to (2.8) with n = 2, they had to impose in addition certain so called residual relations in order to prove for example that the determinant is central. Such relations were also required in [FV96] in a different algebraic setting. In the setting of operator algebras, where the algebras consist of linear operators on a vector space depending meromorphically on the spectral variables, as in [FV97], such relations are consequences of the ordinary RLL-relations by taking residues. Another motivation for our procedure is that h-bialgebroids associated to gauge equivalent R-matrices should be isomorphic. In particular one should be allowed to multiply the R-matrix by any nonzero meromorphic function of the spectral variable without changing the isomorphism class of the associated algebra (for the full definition of gauge equivalent R-matrices see [EV98]).
These considerations suggest the following procedure for constructing an h-bialgebroid from a quantum dynamical R-matrix with spectral parameter.
Let h be a finite-dimensional abelian Lie algebra, V = α∈h * V α a finite-dimensional diagonalizable h-module and R : h * × C × → End h (V ⊗ V ) a meromorphic function. We attach to this data an h-bialgebroid A R as follows. Let {e x } x∈X be a homogeneous basis of V , where X is an index set. The matrix elements R ab xy : They are meromorphic on h * × C × . Define ω : X → h * by e x ∈ V ω(x) . LetÃ R be the complex associative algebra with 1 generated by {L xy (z) : x, y ∈ X, z ∈ C × } and two copies of M h * , whose elements are denoted by f (λ) and f (ρ), respectively, with defining for all x, y ∈ X, z ∈ C × and f ∈ M h * . The bigrading onÃ R is given by L xy (z) ∈ (Ã R ) ω(x),ω(y) for x, y ∈ X, z ∈ C × and f (λ), f (ρ) ∈ (Ã R ) 00 for f ∈ M h * . The moment maps are defined by µ l (f ) = f (λ), µ r (f ) = f (ρ). The counit and comultiplication are defined by This makesÃ R into an h-bialgebroid. Consider the ideal inÃ R generated by the RLL-relations x,y∈X R xy ac (λ, where a, b, c, d ∈ X, and z 1 , z 2 ∈ C × . More precisely, to account for possible singularities of R, we let I R be the ideal inÃ R generated by all relations of the form where a, b, c, d ∈ X, z 1 , z 2 ∈ C × and ϕ : C × → C is a meromorphic function such that the limits exist. We define A R to beÃ R /I R . The bigrading descends to A R because (3.10) is homogeneous, of bidegree ω(a) + ω(c), ω(b) + ω(d), by the h-invariance of R. One checks that ∆(I R ) ⊆Ã R ⊗I R + I R ⊗Ã R and ε(I R ) = 0. Thus A R is an h-bialgebroid with the induced maps.
Remark 3.4. Objects in Felder's tensor category associated to an R-matrix R are certain meromorphic functions L : [F95]. After regularizing L with respect to the spectral parameter it will give rise to a dynamical representation of the h-bialgebroid A R in the same way as in the non-spectral case treated in [EV98]. The residual relations incorporated in (3.10) are crucial for this fact to be true in the present, spectral, case.

Operator form of the RLL relations
It is well-known that the RLL-relations (3.9) can be written as a matrix relation. We show how this is done in the present setting. It will be used later in Section 6.2.
Assume R ab xy (ζ, z) are defined, as meromorphic functions of ζ ∈ h * , for any z ∈ C × . Define R(λ, z), for a, b ∈ X, u ∈ A R . Note that the λ and ρ in the left hand side are not variables but merely indicates which moment map is to be used. For z ∈ C × we also define Here E xy are the matrix units in End(V ) and A R acts on itself by left multiplication. The RLL relation (3.9) is equivalent to This can be seen by acting on e b ⊗ e d ⊗ 1 in both sides of (3.11), and collecting and equating terms of the form e a ⊗ e c ⊗ u. The matrix elements of the R-matrix in the right hand side can then be moved to the left using that R is h-invariant, and relation (3.5).

The algebra F ell (M(n))
We now specialize to the case where h is the Cartan subalgebra of sl(n), V = C n and R is given by (2.8)-(2.10). The case n = 2 was considered in [KNR04]. We will show that (3.10) contains precisely one additional family of relations, as compared to (3.9), and we write down all relations explicitly. When we apply the generalized FRST-construction to this data we obtain an hbialgebroid which we denote by F ell (M (n)). The generators L ij (z) will be denoted by e ij (z). Thus F ell (M (n)) is the unital associative C-algebra generated by e ij (z), i, j ∈ [1, n], z ∈ C × , and two copies of M h * , whose elements are denoted by f (λ) and f (ρ) for f ∈ M h * , subject to the following relations and z ∈ C × , and n x,y=1 R xy ac (λ, which substituted into (4.2) yields four families of relations: e ab (z 1 )e ab (z 2 ) = e ab (z 2 )e ab (z 1 ), (4.4a) where a, b, c, d ∈ [1, n], a = c and b = d. Since θ has zeros precisely at p k , k ∈ Z, α and β have poles at z = q −2 p k , k ∈ Z. Thus (4.4b)-(4.4d) are to hold for z 1 , z 2 ∈ C × with z 1 /z 2 / ∈ {p k q −2 : k ∈ Z}. In (3.10), assuming a = c, b = d, and taking z 1 = z, z 2 = p k q 2 z, ϕ(w) = q −1 θ(q 2 w) qθ(q −2 w) , and using the limit formulas (2.19), we obtain the relation α(λ ac , q 2 ) e ab (z)e cd (p k q 2 z) − q 2kλca e cb (z)e ad (p k q 2 z) = = α(ρ bd , q 2 )e cd (p k q 2 z)e ab (z) − q 2kρ bd β(ρ bd , q 2 )e cb (p k q 2 z)e ad (z). (4.5) This identity does not follow from (4.4a)-(4.4d) in an obvious way. It will be called the residual RLL relation.
Proof. Assume we have a relation of the form (3.10) and that a limit in one of the terms, lim w→z ϕ(w)R ab xy (λ, w), say, exists and is nonzero. Then one of the following cases occurs.
1. At w = z, ϕ(w) and R ab xy (λ, w) are both regular. If this holds for all terms, then the relation is just a multiple of one of (4.4a)-(4.4d).
2. At w = z, ϕ(w) has a pole while R ab xy (λ, w) is regular. Then R ab xy (λ, w) must vanish identically at w = z. The only case where this is possible is when x = y and R ab xy (λ, w) = α(λ xy , w) and z = p k . But then there is another term containing β which is never identically zero for any z, and hence the limit in that term does not exist.
3. At w = z, ϕ(w) is regular while R ab xy (λ, w) has a pole. Since these poles are simple and occur only when z ∈ q −2 p Z , the function ϕ must have a zero of multiplicity one there. We can assume without loss of generality that ϕ has the specific form Then, if a = c and b = d, (3.10) becomes the residual RLL relation (4.5).
If instead c = a, b = d, and we take z 1 = z, z 2 = p k q 2 z in (3.10) we get, using (2.19), 0 = α(ρ bd , p k q 2 )e ad (p k q 2 z)e ab (z) − β(ρ bd , p k q 2 )e ab (p k q 2 z)e ad (z), or, rewritten, However this relation is already derivable from (4.4b) as follows. Take z 1 = p k q 2 z and z 2 = z in (4.4b) and multiply both sides by q 2kρ bd E(ρ bd −1) E(ρ bd +1) and then use (4.4b) on the right hand side.
Similarly to the previous case, this identity follows already from (4.4c).

Corepresentations of h-bialgebroids
We recall the definition of corepresentations of an h-bialgebroid given in [KR01].
Given an h-space V and an h-bialgebroid A, we define A ⊗V to be the h * -graded space Definition 5.3. A left h-comodule algebra V over an h-bialgebroid A is a left corepresentation ∆ V : V → A ⊗V and in addition a C-algebra such that V α V β ⊆ V α+β and such that ∆ V is an algebra morphism, when A ⊗V is given the natural algebra structure.
Right corepresentations and comodule algebras are defined analogously.

The comodule algebras Λ and Λ ′ .
We define in this section an elliptic analog of the exterior algebra, following [KN06], where it was carried out in the trigonometric non-spectral case. It will lead to natural definitions of elliptic minors as certain elements of F ell (M (n)). One difference between this approach and the one in [FV97] is that the elliptic exterior algebra in our setting is really an algebra, and not just a vector space. Another one is that the commutation relations in our elliptic exterior algebras are completely determined by requiring the natural relations (5.2a), (5.2b), (5.3), and that the coaction is an algebra homomorphism. This fact can be seen from the proof of Proposition 5.4. Since the proof does not depend on the particular form of α and β, we can obtain exterior algebras for any hbialgebroid obtain through the generalized FRST-construction from an R-matrix in the same manner. In particular the method is independent of the gauge equivalence class of R.
Let Λ be the unital associative C-algebra generated by v i (z), 1 ≤ i ≤ n, z ∈ C × and a copy of M h * embedded as a subalgebra subject to the relations We require also the residual relation of (5.2c) obtained by multiplying by ϕ(z/w) = q −1 θ(q 2 z/w) qθ(q −2 z/w) and Similarly one proves that (5.2c),(5.2d) are preserved.
Relation (5.2c) is not symmetric under interchange of j and k. We now derive a more explicit, independent, set of relations for Λ. We will use the function E, defined in (2.5).
(ii) It follows from the relations that each monomial in Λ can be uniquely written as It remains to show that the set (5.6) is linearly independent over M h * . Assume that a linear combination of basis elements is zero, and that the sum has minimal number of terms. By multiplying from the right or left by v j (w) for appropriate j, w we can assume the sum is of the form where t is an indeterminate. Formally, consider C(t) ⊗ Λ, the tensor product (over C) of Λ by the field of rational functions in t. We identify Λ with its image under Λ ∋ v → 1 ⊗ v ∈ C(t) ⊗ Λ, and view C(t) ⊗ Λ naturally as a vector space over C(t). By relations (5.5a)-(5.5d), there is a C-algebra automorphism for f ∈ M h * , p ∈ C(t) and i ∈ [1, n], z ∈ C × and extend D to a C-linear map D : The point is that the requirement (5.9) respects relations (5.5a)- By applying D repeatedly we get u 1 (z 1 )+ · · · + u r (z r ) = 0, Inverting the Vandermonde matrix (p j (t) i−1 ) ij we obtain u j (z j ) = 0 for each j, i.e. f j (ζ) = 0 for each j. This proves linear independence of (5.6).
Analogously one defines a right comodule algebra Λ ′ with generators w i (z) and f (ζ) ∈ M h * . The following relations will be used: Λ ′ has also M h * -basis of the form (5.6). In fact Λ and Λ ′ are isomorphic as algebras.

Action of the symmetric group
From (4.4),(4.5) we see that S n × S n acts by C-algebra automorphisms on F ell (M (n)) as follows is given by permutation of coordinates: Also, S n acts on Λ by C-algebra automorphisms via Similarly we define an S n action on Λ ′ .
Lemma 5.6. For each v ∈ Λ, w ∈ Λ ′ and any σ, τ ∈ S n we have Proof. By multiplicativity, it is enough to prove these claims on the generators, which is easy.
6 Elliptic quantum minors 6.1 Definition and define the left and right elliptic sign functions for σ ∈ S n . In fact, E(ζ ij )/E(ζ ji ) = −1 so sgn [1,n] (σ; ζ) is just the usual sign sgn(σ). However we view this as a "coincidence" depending on the particular choice of R-matrix from its gauge equivalence class. We keep our notation to emphasize that the methods do not depend on this choice of R-matrix.
We will denote the elements of a subset I ⊆ [1, n] by i 1 < i 2 < · · · .
Proof. We prove (6.4). The proof of (6.5) is analogous. We proceed by induction on #I = d, the case d = 1 being clear. If d > 1, set By the induction hypothesis, the left hand side of (6.4) equals Now v σ(i d ) (q 2(d−1) z) commutes with sgn I ′ (σ, ζ) since the latter only involves ζ ij with i, j = σ(i d ). Using the commutation relations (5.5b) we obtain Introduce the normalized monomials Corollary 6.2. Let I ⊆ [1, n]. For any permutation σ ∈ S n , σ(v I (z)) = v σ(I) (z), σ(w I (z)) = w σ(I) (z), (6.10) for any z ∈ C × . In particular v I (z) and w I (z) are fixed by any permutation which preserves the subset I.
Proof. Let J = σ(I). Then The proof for w I (z) is analogous.
We are now ready to define certain elements of the h-bialgebroid F ell (M (n)) which are analogs of minors. − → ξ J I (z) respectively, can be defined by where the sums are taken over all subsets of [1, n].
for any σ ∈ S I , and (6.14) for any τ ∈ S J . Moreover, for any (σ, τ ) ∈ S n × S n and z ∈ C × .
Remark 6.4. In Theorem 6.10 we will prove that, in fact, We prove the statements concerning the left elliptic minor Thus (6.11) holds when ← − ξ J I (z) is defined by (6.13) with σ = Id. Then the right hand side of (6.13) equals (σ, Id)( ← − ξ J I (z)). Thus only (6.15) remains. Using (5.12) and Corollary 6.2 we have On the other hand, again by Corollary 6.2, where we made the substitution J → τ (J). This proves the first equality in (6.15). The statements concerning the right elliptic minors are proved analogously.

Equality of left and right minors
The goal of this section is to prove Theorem 6.10 stating that the left and right elliptic minors coincide. We use ideas from Section 3 of [FV97], where the authors study the objects of Felder's tensor category [F95] and associate a linear operator (product of Rmatrices) on V ⊗n to each diagram of a certain form, a kind of braid group representation. Then they consider the operator associated to the longest permutation, in [ZSY03] called the Cherednik operator. Instead of working with representations, we proceed inside the h-bialgebroid F ell (M (n)) and consider certain operators on V ⊗n ⊗ F ell (M (n)) depending on n spectral parameters. Using the analog of the Cherednik operator we prove an extended RLL-relation (6.25). Theorem 6.10 then follows by extracting matrix elements from both sides of this matrix equation.
In this section, we set F = F ell (M (n)). Recall the operators from Section 3.3, defined for any h-bialgebroid A R obtained from the FRST-construction. When specializing to F we get operators R(λ, z), R(ρ, z) ∈ End(V ⊗ V ⊗ F), where V = C n . For z ∈ C × , define the following linear operators on V ⊗n ⊗ F: The upper indices in parenthesis are tensor leg numbering and indicate the tensor factors the operator should act on. The limits are taken in the sense of taking limits of each matrix element. These operators are well-defined for any z, since we multiply away the singularities in z of α and β (2.9),(2.10). Let E n denote the algebra of all functions The symmetric group S n acts on E n by for F (z) ∈ E n and σ ∈ S n . In the right hand side of (6.16), σ acts on (C × ) n by permuting coordinates, and on V ⊗n by permuting the tensor factors. For example we have (23) R 12 (λ, z 1 /z 2 ) = R 13 (λ, z 1 /z 3 ).
Consider the skew group algebra E n * S n , defined as the algebra with underlying space E n ⊗ CS n , where CS n is the group algebra, with the multiplication for σ, τ ∈ S n , F (z), G(z) ∈ E n . Since σ acts on E n by automorphisms, E n * S n is an associative algebra. The constant function z → Id V ⊗n ⊗F ⊗ (1) is the unit element. Let B n be the monoid (set with unital associative multiplication) generated by {s 1 , . . . , s n−1 } and relations Let σ i = (i i + 1) ∈ S n . We have an epimorphism π : B n → S n given by π(s i ) = σ i , π(1) = (1). Define Here and below we use h ≥k to denote the expression n j=k h j .
Proposition 6.5. W extends to a well-defined morphism of monoids, i.e. a map Proof. We have to show the relations Relation (6.18) follows from the QDYBE (2.6). For example, W (s i )W (s i+1 )W (s i ) equals Relation (6.19) is easy to check, using the h-invariance of R. (6.20) From this and the product rule (6.17) follows that for b 1 , b 2 ∈ B n . By replacing λ by ρ we get similarly operators W b (ρ, z).
Recall the operators L(z) ∈ End(V ⊗F) from Section 3.3. Define for z ∈ C × , i ∈ [1, n], If i, j, k are distinct, then one can check that Due to the RLL relations (3.10) we have Let τ d be the image of t d in S n : Proposition 6.6. Let 1 ≤ d ≤ n. For any z = (z 1 , . . . , z d ) ∈ (C × ) d we have Proof. We use induction on d. The case d = 1 is trivial, while d = 2 is the RLL relation (6.24). If d > 2, write t d = t d−1 u d , where u d = s d−1 s d−2 · · · s 1 . Thus, by (6.21), We claim that where h [a,b] means a≤j≤b h j . Thus (6.28) Using (6.22) and the RLL relation (6.24) repeatedly, we obtain (6.27). Now the proposition follows by induction on d, using that which follows from (6.22).
Lemma 6.8. Fix 2 ≤ d ≤ n and i < d. Then there are elements b, c ∈ B n such that t d = s i b and t d = cs i .
Proof. Since t 2 = s 1 and t 3 = s 1 s 2 s 1 = s 2 s 1 s 2 , the statement clearly holds for d = 2, 3. Assuming d > 3, we first prove the existence of b. − 1 rightmost factors s d−1 , . . . , s 1 as far to the left as possible, using that s j s k = s k s j when |j − k| > 1. This gives Then use s j s j+1 s j = s j+1 s j s j+1 repeatedly, working from right to left, to obtain Finally, s d−1 can be moved to the left of t d−2 since the latter is a product of s j 's with j ≤ d − 3.
To prove the existence of c we note that B n carries an involution * : B n → B n satisfying (a 1 a 2 ) * = a * 2 a * 1 for any a 1 , a 2 ∈ B n , defined by s * j = s j for j ∈ [1, n] and 1 * = 1. Thus it suffices to show that t * d = t d for any d. This is trivial for d = 2, 3. When d > 3 we have, by induction on d, Proposition 6.9. Let w = (z 0 , q 2 z 0 , . . . , q 2(n−1) z 0 ), where z 0 = 0 is arbitrary, and let σ, τ ∈ S n . Then From this and the definitions of the sign functions, (6.2)-(6.3), the claims follow. Next, we prove (6.32) by induction on the sum ℓ of the lengths of σ and τ . If ℓ = 0 it is trivial. Assuming (6.32) holds for (σ, τ ) we prove it holds for (σσ i , τ ) and (σ, τ σ i ) where i is arbitrary. Let i ∈ [1, n]. By Lemma 6.8 we have t n = s i b for some b ∈ B n . We have x 1 ,...,xn .
Theorem 6.10. For any subsets I, J ⊆ [1, n] and z ∈ C × , the left and right elliptic minors coincide: . We denote this common element by ξ J I (z).
Theorem 6.11. (i) Let I 1 , I 2 , J ⊆ [1, n] and set I = I 1 ∪ I 2 . Then Proof. We have On the other hand, Equating these expressions proves (6.38) since, by Proposition 5.5, the set {v J (z) : J ⊆ [1, n]} is linearly independent over M h * . The second part is completely analogous, using the right comodule algebra Λ ′ in place of Λ.
In Section 7.4 we will need the following lemma, relating the left and right signums S l (I, J; ζ) and S r (I, J; ζ), defined in (6.36),(6.37). In the non-spectral trigonometric case the corresponding identity was proved in [N05] (in proof of Proposition 4.1.22). where ω(I) = i∈I ω(i).
Proof. First we claim that, we have the following explicit formulas: S l (I, J; ζ) = i∈I,j∈J E(ζ ji + 1), (6.41) Recall the definition, (6.8), of v I (z). Since E is odd, relation (5.5b) implies that Also, F J (ζ) only involves ζ ij with i, j ∈ J so it commutes with any v k (z) with k ∈ I (since I ∩ J = ∅). From these facts we obtain where K = (I × J) ∪ (J × I). This proves (6.41). Similarly one proves (6.42). Now we have Here we used that for any i ∈ I, j ∈ J we have ω(J)(E ii ) = 0 and ω(J)(E jj ) = 1 and hence (ω(J)) ij = −1.
7 The cobraiding and the elliptic determinant

Cobraidings for h-bialgebroids
The following definition of a cobraiding was given in [R04]. When h = 0 the notion reduces to ordinary cobraidings for bialgebras.
Definition 7.1. A cobraiding on an h-bialgebroid A is a C-bilinear map ·, · : A × A → D h such that, for any a, b, c ∈ A and f ∈ M h * , The following definition was given in unpublished notes by Rosengren [R07]. The terminology is motivated by Proposition 7.6 below concerning FRST algebras A R , but it makes sense for arbitrary h-bialgebroids.
for all a, b ∈ A.
7.2 Cobraidings for the FRST-algebras A R .
be a meromorphic function and let A R be the h-bialgebroid associated to R as in Section 3.2.
Proposition 7.3. Assume that ϕ : C × → C is a holomorphic function, not vanishing identically, such that, for each x, y, a, b ∈ X, z ∈ C × , the limit lim w→z ϕ(w)R ab xy (ζ, w) exists and defines a meromorphic function in M h * . Then the following statements are equivalent: (ii) R satisfies the QDYBE (2.6).
Remark 7.4. a) The identity (7.1g) is not necessary when proving that (i) implies (ii). Without assuming (7.1g), ·, · is a pairing on A cop × A. See [R04]. b) Without the factor ϕ(w), the cobraiding is not well-defined if R(ζ, z) has poles in the z variable. We also remark that the residual relations (3.10) are necessary for (ii) to imply (i).
Proof. The proof is straightforward and is carried out in [N05], Lemma 2.2.5, under the assumption that the R-matrix is regular in the spectral variable.
We will now generalize slightly the notion of a unitary cobraiding on A R to account for spectral singularities in the R-matrix as follows.
Call a ∈ A R spectrally homogenous if there exist k ∈ Z ≥0 and z 1 , . . . , The multiset {z i } i is called the spectral degree of a and is denoted sdeg(a). Note that the spectral degree of a nonzero spectrally homogenous element is uniquely defined, since the RLL-relations (3.10) are spectrally homogenous.
Let ϕ : C × → C be holomorphic. For spectrally homogenous elements a, b ∈ A R , define the regularizing factor ϕ(a, b) by Definition 7.5. Let ϕ : for all spectrally homogenous a, b ∈ A R .
The following proposition was proved in [R07] if the spectral variables is taken to be generic so that no regularizing factors are needed.
Proposition 7.6. Suppose R : h * × C × → End C (V ⊗ V ) satisfies the QDYBE and is unitary: R(ζ, z)R(ζ, z −1 ) (21) = Id V ⊗V . Suppose ϕ : C × → C is nonzero holomorphic such that lim w→z ϕ(w)R ab xy (ζ, w) exists and is a holomorphic function in M h * . Then the cobraiding ·, · on A R given in Proposition 7.3 is unitary with respect to ϕ.
Proof. Since both sides are holomorphic in the spectral variables, it is enough to prove it for generic values. We claim that for such values, ϕ(a, b) a, b R = a, b where ·, · R is the cobraiding, defined only for generic spectral values, determined by L ij (z), L kl (w) R = R jl ik (ζ, z/w)T −ω(i)−ω(k) . Indeed, this claim follows by induction from the identities (7.1d),(7.1e) using that ϕ(a 1 , b) ϕ(a 2 , c) = ϕ(a 3 , bc) and ϕ(c, a 1 ) ϕ(b, a 2 ) = ϕ(cb, a 3 ) for spectrally homogenous a i , b, c ∈ A R , the a i having the same spectral degree.
Since the R-matrix R is unitary, the statement of the lemma now follows from the identity holding for generic spectral values which was proved by Rosengren [R07].

The case of F ell (M (n))
Specializing further to the algebra of interest, F ell (M (n)), we obtain the following corollary.

Properties of the elliptic determinant
A common method used to study quantum minors and prove that quantum determinants are central is the fusion procedure, going back to work by Kulish and Sklyanin [KS82]. Another approach, using representation theory, was developed by Noumi, Yamada and Mimachi [NYM]. In this section we show how to prove that the elliptic determinant is central using the properties of the cobraiding on F ell (M (n)) and how to resolve technical issues connected with the spectral singularities of the elliptic R-matrix. Let A = F ell (M (n)). When I = J = [1, n] we set det(z) = ξ J I (z) (7.10) for z ∈ C × , where ξ J I (z) is the elliptic minor given in Theorem 6.10. Thus one possible expression for det(z) is e σ(1)1 (z)e σ(2)2 (q 2 z) · · · e σ(n)n (q 2(n−1) z).
(7.11) From the coassociativiy and counity axioms for a corepresentation follows that det(z) is grouplike, proving part a). The rest of this section is devoted to the proof of part b). It follows from the definition that det(z) ∈ A 00 and thus it commutes with f (ρ) and f (λ) for any f ∈ M h * . To prove that it commutes with the generators e ij (z) we need several lemmas which we now state and prove.
If q 2k w/z / ∈ p Z for any k ∈ {0, 1, . . . , n − 2}, this follows from the cobraiding identity (7.1g) with a = det(w), b = e 11 (z) by dividing by the nonzero number n−2 k=0 θ(q k w/z). So assume q 2k w/z ∈ p Z for some k ∈ {0, 1, . . . , n − 2}. We again use the iterated Laplace expansion (7.27). For simplicity of notation, we write it as det(w) = a 1 a 2 a 3 where a 2 is the 2×2 minor. Put b = e 11 (z). Substituting this, and expanding a 1 a 2 a 3 , b ′ using (7.1d), we get after simplification Now using the cobraiding identity (7.1g) and its primed version for quadratic minors (7.24), we can move the b all the way to the left. Doing the steps backwards the claim follows.
The factors involving the dynamical variable ζ cancel and the claim follows. By Lemma 7.11 b) and Lemma 7.12 we conclude that det(w) commutes with e 11 (z) if q 2n w/z / ∈ p Z . By applying an automorphism from the S n × S n -action on A as defined in Section 5.3 and using that det(z) is fixed by those, by relation (6.15), we conclude that det(w) commutes with any e ij (z) as long as q 2n w/z / ∈ p Z . For the remaining case we can note that relations (4.4),(4.5) imply that there is a C-linear map T : F ell (M (n)) → F ell (M (n)) such that T (ab) = T (b)T (a) for all a, b ∈ F ell (M (n)), given by T e ij (z) = e ij (z −1 ), T f (λ) = f (−λ), T f (ρ) = f (−ρ), for all f ∈ M h * , i, j ∈ [1, n] and z ∈ C × . One verifies that T (det(z)) = det(q −2(n−1) z −1 ).

The antipode
We use the following definition for the antipode, given in [KR01]. where m denotes the multiplication and ε(a)1 is the result of applying the difference operator ε(a) to the constant function 1 ∈ M h * .

Concluding remarks
To define the antipode we only needed that e ij (z) commutes with det(q −2(n−1) z). This can also be proved using the Laplace expansions. Perhaps one could avoid problems with spectral poles and zeros of the R-matrix by thinking of the algebra as generated by meromorphic sections of a M h * ⊕h * -line bundle over the elliptic curve C × /{z ∼ pz}. In this direction we found that the relation e ij (pz) = q λ i −ρ j e ij (z) respects the RLL-relation (here h should be the Cartan subalgebra of gl n ). This relation should then most likely be added to the algebra.
It would be interesting to develop harmonic analysis for the elliptic GL(n) quantum group, similarly to [KR01]. In this context it is valuable to have an abstract algebra to work with, and not only a tensor category analogous to a category of representations. For example the analog of the Haar measure seems most naturally defined as a certain linear functional on the algebra.