The dynamical U(n) quantum group

We study the dynamical analogue of the matrix algebra M(n), constructed from a dynamical R-matrix given by Etingof and Varchenko. A left and a right corepresentation of this algebra, which can be seen as analogues of the exterior algebra representation, are defined and this defines dynamical quantum minor determinants as the matrix elements of these corepresentations. These elements are studied in more detail, especially the action of the comultiplication and Laplace expansions. Using the Laplace expansions we can prove that the dynamical quantum determinant is almost central, and adjoining an inverse the antipode can be defined. This results in the dynamical GL(n) quantum group associated to the dynamical R-matrix. We study a *-structure leading to the dynamical U(n) quantum group, and we obtain results for the canonical pairing arising from the R-matrix.


Introduction
Dynamical quantum groups have been introduced recently by Etingof and Varchenko [11], see the review paper by Etingof and Schiffmann [9] for an overview and references to the literature, and related algebraic structures have been studied by Lu [23], Xu [35] in the context of deformations of Poisson groupoids, and by Takeuchi [34]. Brzeziǹski and Militaru [4] compare the various constructions of [23], [35], [34]. In this paper we stick to the definition of Etingof and Varchenko [11] with a slight modification as in [19]. In order to keep the paper self-contained as much as possible we recall the definition in section 2. We also recall the FRST-construction associated to a solution of the dynamical R-matrix, which gives a wealth of examples, and which we consider explicitly for the trigonometric R-matrix in the gl(n)-case.
It is well-known that quantum groups have a natural link with special functions of basic hypergeometric type, and in [19] it is shown that this remains valid for the simplest example of a dynamical quantum group associated to the trigonometric R-matrix for SL (2) and in [18], see also [17], for the elliptic R-matrix for SL(2) giving a dynamical quantum group theoretic interpretation of elliptic hypergeometric series. In particular, [19] gives a dynamical quantum group theoretic interpretation of Askey-Wilson and q-Racah polynomials having many similarities to the interpretation of these polynomials on the (ordinary) quantum SL(2) group using the twisted primitive elements as introduced by Koornwinder [21], see also [28], [15]. This naturally suggests a link between these two approaches, and the link is established by Stokman [32] using the coboundary element of Babelon, Bernard and Billey [2], a universal element in the tensor product of the quantized universal algebra. This element also has a natural interpretation in the context of twisted primitive elements as shown by Rosengren [30]. However, the coboundary element is only known for the sl(2)-case, but there are conjectures about its form for the sl(n) case, see Buffenoir and Roche [5]. The notion of twisted primitive elements of Koornwinder [21], and especially its generalization to co-ideals, has turned out to be enormously fruitful for the interpretation of special functions of one or many variables as spherical functions on quantum groups or quantum symmetric spaces, see e.g. [6], [8], [22], [27], [33].
As one of the highlights of the application of Lie theory to special functions we mention the group theoretic derivation of the addition formula for Jacobi polynomials as obtained by Koornwinder [20] by working on the symmetric space U (n)/U (n − 1) and establishing the spherical and associated spherical elements in terms of Jacobi polynomials, see also Askey [1,Lecture 4] for a nice introduction. In the case n = 2 this gives the addition formula for Legendre polynomials. For q-analogues of addition formulas for the Legendre polynomials see the overview [16]. In the quantum group setting, Noumi, Yamada and Mimachi [29] established the little q-Jacobi polynomials as spherical functions, and Floris [13] calculated the associated spherical elements in terms of little q-Jacobi polynomials and derived an addition formula. On the other hand, using the notion of co-ideals, Dijkhuizen and Noumi [8] established Askey-Wilson polynomials as spherical functions on a quantum analogue of U (n)/U (n − 1).
In light of the above it is natural to ask for the spherical (and associated spherical) elements on the dynamical quantum group analogue of U (n)/U (n−1), and if a precise link to special functions can be established. For this we need to study the dynamical U (n) quantum group more closely, and in a previous paper [17] we have studied general aspects of dynamical quantum groups for this purpose. In case n = 2 [19] shows that the algebraic approach to quantum groups as discussed by Dijkhuizen and Koornwinder [7] is applicable, and we expect this to hold true for general n. This paper serves as a first step in this specific programme by defining the dynamical U (n) quantum group and studying some of its elementary properties. In a future paper its corepresentation theory and (associated) spherical functions have to be studied.
The general theory provides us with a dynamical analogue of the algebra of functions on the space of n × n-matrices. In section 2 we recall the algebraic notions and FRST-construction of Etingof and Varchenko [10], [11], see also [9], and this gives an explicit presentation by generators and relations for this dynamical analogue. In order to make this into a dynamical quantum group, we need to equip (a suitable extension) of this algebra with an antipode. For this purpose we study the dynamical analogues of the minor determinants, which are introduced as matrix elements of corepresentations which are analogues of the natural representation in the exterior algebra. There exist a left and right corepresentation, and we show that the matrix elements, i.e. the dynamical quantum minor determinants, are equal using an identity for Hall-Littlewood polynomials. In particular, this gives a dynamical quantum determinant. This is done in section 3. In section 4, we continue the study of these dynamical quantum minor elements and we discuss the appropriate analogues of the Laplace expansions. In section 5 we show how the Laplace expansions imply that the dynamical quantum determinant is almost central, and localizing we find the dynamical GL(n) quantum group for which we give an explicit expression for the antipode. The treatment of dynamical quantum minor elements, the Laplace expansions, and the extension to a h-Hopf algebroid is very much motivated by the paper by Noumi, Yamada and Mimachi [29]. We also introduce a * -structure, so that we obtain the dynamical U (n) quantum group in section 6. Finally, in section 7 we study the natural pairing, as introduced by Rosengren [31], see also [17], for the case of the dynamical GL(n) quantum group and the dynamical U (n) quantum group.
Acknowledgement. We thank Hjalmar Rosengren and Jasper Stokman for useful discussions.
2. The dynamical analogue of the matrix algebra M (n) In this section we give the general definitions of the theory of dynamical quantum groups and we recall the generalized FRST-construction. To define the h-bialgebroid F R (M (n)) we apply this construction to a solution of the quantum dynamical Yang-Baxter equation (QDYBE).
Let h be a finite dimensional complex vector space, viewed as a commutative Lie algebra, with dual space h * . Let V = α∈h * V α be a diagonalizable h-module. The quantum dynamical Yang-Baxter equation is given by (2.1) is a meromorphic function, h indicates the action of h and the upper indices are leg-numbering notation for the tensor product. For instance, In the example we study, we identify h ∼ = h * ∼ = C n and take V an n-dimensional vector space with basis {v 1 , . . . , v n }. The R-matrix R : h * → End h (V ⊗ V ) we consider is given by where λ = (λ 1 , . . . , λ n ), E ab ∈ End(V ) such that E ab v c = δ bc v a and the meromorphic functions h 0 and g are given by Etingof and Varchenko [11] obtain this R-matrix as the exchange matrix for the vector representation of GL(n).
2.1. h-Hopf algebroids and the generalized FRST-construction. We recall the definition of h-Hopf algebroids, the algebraic notion for a dynamical quantum groups, and the generalized FRST-construction. Let h be a finite dimensional complex vector space, with dual space h * . Denote by M h * the field of meromorphic functions on h * . For α ∈ h * we denote by T α : A morphism of h-algebras is an algebra homomorphism which preserves the bigrading and the moment maps.
Let A and B be two h-algebras. The matrix tensor product A⊗B is the h * -bigraded vector space (2.4) The multiplication (a ⊗ b)(c ⊗ d) = ac ⊗ bd for a, c ∈ A and b, d ∈ B and the moment maps µ l (f ) = µ A l (f ) ⊗ 1 and µ r (f ) = 1 ⊗ µ B r (f ) make A⊗B into a h-algebra.
Example. Let D h * be the algebra of difference operators acting on M h * , consisting of the operators i f i T β i , with f i ∈ M h * and β i ∈ h * . This is a h-algebra with the bigrading defined by f T −β ∈ (D h * ) ββ and both moment maps equal to the natural embedding. For any h-algebra A, there are canonical isomorphisms A ∼ = A⊗D h * ∼ = D h * ⊗A, defined by The algebra D h * plays the role of the unit object in the category of h-algebras.

Definition 2.2.
A h-bialgebroid is a h-algebra A equipped with two h-algebra homomorphisms ∆ : A → A⊗A (the comultiplication) and ε : For the definition of the antipode we follow [19]. If there exists an antipode on a h-bialgebroid, it is unique. Furthermore, the antipode is antimultiplicative, anti-comultiplicative, unital, counital and interchanges the moment maps µ l and µ r , see [19,Prop. 2.2]. In Definition 2.3 the maps m • (Id ⊗ S) and m • (S ⊗ Id) are well-defined on A⊗A, see [17].
Example. (i) We can equip D h * with a h-Hopf algebroid structure with comultiplication ∆ : D h * → D h * ⊗D h * ∼ = D h * the canonical isomorphism, counit ε : D h * → D h * the identity and antipode defined by S(f T α ) = T −α • f . (ii) For a h-Hopf algebroid A with invertible antipode, the opposite and co-opposite are also h-Hopf algebroids. The opposite algebra A opp is the algebra A with opposite multiplication. Then we equip A opp with a h-Hopf algebroid structure by defining ( The co-opposite algebra A cop has the same algebra structure but µ cop Let λ → λ be a complex conjugation on h * , and denote f (λ) = f (λ) for all f ∈ M h * .
Until this point we have seen only the example D h * of a h-bialgebroid. The generalized FRSTconstruction provides many examples of h-bialgebroids from R-matrices, see [10], [9], [12], [19]. We recall the construction and we apply the construction to the R-matrix in (2.2) to obtain the main object of study for this paper.
Let h and M h * be as before, V = α∈h * V α be a finite-dimensional diagonalizable h-module and R : h * → End h (V ⊗ V ) a meromorphic function that commutes with the h-action on V ⊗ V .
Let {v x } x∈X be a homogeneous basis of V , where X is an index set. Write R ab xy (λ) for the matrix elements of R, and define ω : . Let A R be the unital complex associative algebra generated by the elements {L xy } x,y∈X together with two copies of M h * , embedded as subalgebras. The elements of these two copies will be denoted by f (λ) and f (µ), respectively. The defining relations of A R are f (λ)g(µ) = g(µ)f (λ), f (λ)L xy = L xy f (λ + ω(x)) and f (µ)L xy = L xy f (µ + ω(y)) for all f , g ∈ M h * , together with the RLL-relations The hinvariance of R ensures that the bigrading is compatible with the RLL-relations (2.7). Finally the counit and comultiplication defined by equip A R with the structure of a h-bialgebroid, see [10].
Remark 2.6. The case n = 2 and restricting to functions depending only on λ 1 − λ 2 gives back the case studied in [19].
As in [29] for the quantum case and in [19] for n = 2, we can give a linear basis for F R (M (n)). The proof is more involved since we use relations for the functions h, g. Proposition 2.7 is stated for later reference. 11 11 t a 12 12 · · · t a 1n 1n t a 21 21 · · · t ann nn . Then Proof. This follows from the diamond lemma, see [3]. First we introduce a total ordering ≺ by we use the lexicographical ordering on (a 11 , a 12 , . . . , a 1n , a 21 , . . . a 2n , a 31 , . . . , a nn ).
We have the following reduction system, which is compatible with the introduced total order. Assume i < j, k < l, To simplify the coefficients on the right hand side we use ). If we prove that the reduction system is resolvable, the lemma follows from [3, Thm 2.1]. There are 24 types of configuration to be checked. The proof is straightforward using for all λ, µ, ν ∈ h * .

Exterior corepresentations and dynamical quantum minor determinants
We continue with the study of some elementary corepresentations of F R (M (n)) analogous to the action of M (n) on the exterior algebra of C n . Using these corepresentations we find the dynamical determinant in F R (M (n)). First we recall the general definition of a corepresentation of a h-bialgebroid on a h-space, see [19]. We introduce the notion of h-comodule algebras.
In case we want to emphasize the dependence on V we also write f v = µ V (f )v. We next define the tensor product of a h-bialgebroid A and a h- The first equality is in the sense of the natural isomorphism (V⊗A)⊗A ∼ = V⊗(A⊗A) and in the second identity we use the identification If, moreover, V is a unital algebra, we require R (L) to be unital.
So R preserves the relation in (i). Recall that by the M h * -linearity of a corepresentation we have Now we define the h-space W on which we construct a right corepresentation of F R (M (n)). W can be seen as the dynamical analogue of the exterior algebra representation.
Definition 3.5. Let W be the unital associative algebra generated by the elements w i , i ∈ {1, 2, . . . , n} and a copy of M h * embedded as a subalgebra, its elements denoted by f (λ), subject to the relations For an ordered subset . . , n} we use the convention w I = w i 1 · · · w ir , unless mentioned otherwise. Moreover, ∅ is an ordered subset and w ∅ = 1 corresponding to the case r = 0. The following lemma is easily proved. Lemma 3.6. dim M h * W = 2 n and a basis for W is given by {w I : Proof. It is clear that W satisfies the conditions of Definition 3.3(i). To see that R can be extended uniquely to an algebra homomorphism we need to verify where we use the second relation of (2.12) in the last equality. Let us emphasize that the function Similarly we obtain that the relation Using Lemma 3.6 it remains to prove that For I and J ordered subsets with #I = #J we define the elements ξ I J as the corresponding matrix elements; R(w J ) = We call the matrix elements ξ I J the dynamical quantum minor determinants of F R (M (n)) with respect to the subsets I and J. The element ξ {1,...,n} {1,...,n} is called the determinant of F R (M (n)), and is also denoted by det.
This right corepresentation has a left analogue, a left h-comodule algebra V for F R (M (n)). The proofs are analogous to the ones for the right h-comodule algebra W , and are skipped. Definition 3.9. Let V be the unital associative algebra generated by the elements v i , i ∈ {1, . . . , n} and a copy of M h * embedded as a subalgebra, its elements denoted by f (λ), subject to the relations For an ordered subset I = {i 1 , . . . i r } with 1 ≤ i 1 < . . . < i r ≤ n we denote by v I the ordered element v I = v ir · · · v i 1 ∈ V . Let us emphasize that an element v I ∈ V has reversed order compared to w I ∈ W by notational convention.
We call the matrix elements η I J the dynamical quantum minor determinants of F R (M (n)) with respect to the subsets I and J. In Proposition 3.17 we prove that the dynamical quantum minor determinants related to the right and left corepresentation are equal, so we can speak of the dynamical quantum minor determinants of F R (M (n)), without mentioning right or left. First we compute an explicit expression of the dynamical quantum minor determinants which we use in the proof.
Lemma 3.13. For any permutation σ ∈ S r we have the following relation in W ; Proof. We prove by induction on r, for r = 2 and σ = Id it is trivial. If σ = (12) it is just (3.1) for j = i σ(1) , i = i σ (2) . Denote by I ′ the ordered subset of I defined by I \ {i σ(1) }, then Using Lemma 3.13 we calculate the action of the corepresentation R on w j 1 . . . w jr for an arbitrary unordered set {j 1 , . . . , j r }. Then and there is only a non-zero contribution in the right hand side of (3.6) if all k i = k j for i = j. Let I = {i 1 , . . . , i r } be ordered, then we see that the contribution on the right hand side of (3.6) containing the basis element w I in the first leg of the tensor product is given for those terms for which {i 1 , . . . , i r } = {k 1 , . . . , k r } as unordered sets. So there exists for each non-zero term in (3.6) contributing to the term containing w I in the first leg of the tensor product precisely one permutation σ ∈ S r such that k p = i σ(p) . So the term containing w I in the first leg of the tensor product equals Proposition 3.14. Let J be ordered with r = #J, then R(w J ) = #I=#J w I ⊗ ξ I J with the dynamical quantum minor determinants given by for any ρ ∈ S r .
Proof. By Lemma 3.13 and the discussion preceding this proposition we obtain So, the proposition follows from Lemma 3.6.

. t σ(n)ρ(n)
Analogously we obtain an explicit formula for the matrix elements η I J of L. We need to define another generalized sign functionS depending on an ordered subset I, #I = r, and a permutation σ ∈ S r ;S (σ, I)(λ) := where we use h(−λ) = 1/h(λ + 1) for the last equality. Analogous to Lemma 3.13, we have for any permutation σ ∈ S r the following relation in V where I = {i 1 , . . . , i r } is an ordered subset and v I = v ir · · · v i 1 . We get the analogous statement of Proposition 3.14.
Proposition 3.16. Let I = {i 1 , . . . , i r } be an ordered subset, then L(v I ) = #J=#I η I J ⊗ v J with the dynamical quantum minor determinants given by, for any ρ ∈ S r , We now relate the two sets of dynamical quantum minor determinants. For this we need the following identity; σ∈Sr i<j for r indeterminates x 1 , . . . x r . This identity can be found in Macdonald [24, III.1, (1.4)] as the identity expressing that the Hall-Littlewood polynomials for the zero partition gives 1.
Proof. The proof is based on the expressions (3.7) and (3.8), which give the possibility to write a suitable multiple of ξ I J as a double sum over S r , which, by interchanging summations, gives a multiple of η I J . The multiples turn out to be equal. The details are as follows. First we rewrite η I J . Define the longest element σ 0 ∈ S r by σ 0 = 1 2 ... r r r−1 ... 1 . By substituting ρ → ρσ 0 and σ → σσ 0 in (3.8) we get for any ρ ∈ S r . Using this expression for η I J and (3.7) we compute  So it suffices to prove that A(I)(λ) := ρ∈Sr k<l −h(λ i ρ(k) − λ i ρ(l) ) is independent of λ and I: using the explicit expression (2.10) for h and (3.9).

Laplace expansions
In this section, we prove some expansion formulas for the dynamical quantum minor determinants, which are used in the following section to introduce the antipode.
For I 1 , I 2 disjoint ordered subsets of {1, . . . , n}, denote by sign(I 1 ; I 2 ) the element of M h * defined by Then w I 1 w I 2 = µ W (sign(I 1 ; I 2 ))w I if I 1 ∩ I 2 = ∅ and I 1 ∪ I 2 = I. If I 1 ∩ I 2 = ∅ then w I 1 w I 2 = 0 and in this case we define sign(I 1 ; I 2 )(λ) = 0. For I 1 ∩ I 2 = ∅ and I = I 1 ∪ I 2 as ordered subset we have sign(I 1 ; I 2 ) = S(σ, I) where σ is the permutation which maps I 1 ∪ I 2 to the ordered subset I.
where the summation runs over all partitions I 1 ∪ I 2 = I of I such that #I 1 = #J 1 , #I 2 = #J 2 . (ii) The second relation of (4.1) can be rewritten as In the special case #I = #J = n and either J 1 or J 2 contains one element, we get the following expansion formulas for the determinant element. These expansions can be seen as dynamical equivalent of the cofactor expansion across a row or column of the determinant of a matrix.

The dynamical GL(n) quantum group
In this section we extend F R (M (n)) by adjoining an inverse of the determinant. The resulting h-bialgebroid F R (GL(n)) is equipped with an antipode, so it is a h-Hopf algebroid. Proof. Denote by T the n × n-matrix with elements t ij , where i indicates the row index. Using the notation denote byT the n × n-matrix with elements T i j where i indicates the column index. Then the third relation of Corollary 4.3 impliesT T = det I as n 2 identities in F R (M (n)), where I is the n × n-identity matrix. So det T = TT T = T det which implies that det commutes with all generators t ij . Since det ∈ F R (M (n)) 1,1 , we see that det commutes with all elements in M h * that only depend on differences λ i − λ j . By (3.7), det also commutes with ξ I J for all subsets I, J. The last statements follow from Corollary 3.8.
So the determinant element commutes with all generators t ij , but since det ∈ F R (M (n)) 1,1 the element det is not central. However, the set S = {det k } k≥1 satisfies the Ore condition, and this implies that we can localize at det, see [25]. We adjoin F R (M (n)) with the formal inverse det −1 , adding the relations detdet . We denote the resulting algebra by F R (GL(n)) and equip it with a bigrading F R (GL(n)) = m,p∈Z n F R (GL(n)) mp by det −1 ∈ (F R (GL(n))) −1,−1 . Lemma 5.1 implies that det −1 commutes with all dynamical quantum minor determinants ξ I J . By extending the comultiplication and counit of Definition 2.5 by ∆(det −1 ) = det −1 ⊗ det −1 , ε(det −1 ) = T 1 , F R (GL(n)) it is easily checked that F R (GL(n)) is a h-bialgebroid.

2)
and extended as an algebra anti-homomorphism.
Proof. By [19,Prop. 2.2] it suffices to check that S is well-defined and that (2.6) holds on the generators. It is straightforward to check that S preserves the relations (2.11). To see that S preserves the RLL-relations, we apply the antipode to the RLL-relations (2.7). Using (5.1) this gives which is equivalent to We have to prove that (5.4) holds in F R (GL(n)). To show this, we multiply the RLL-relations (2.7) by T k d T l b from the right and by T a j T c i from the left and sum over all a, b, c and d we get, using Corollary 4.3, Multiplying this equation from the left and from the right by det −2 gives (5.4) by the h-invariance of the R-matrix, so S preserves the RLL-relations. From the proof of Lemma 5.1 it follows that S(T )T = T S(T ) = I, where T is defined as in the proof of Lemma 5.1, so (2.6) holds for all generators t ij . The proof of [19,Prop. 2.2] shows that if (2.6) holds for a and b, then it holds for ab, so that in particular (2.6) holds for det. By Lemma 5.1 we find S(det)det = 1 = detS(det), so that S(det) = det −1 . An independent proof of this statement is given in Proposition 5.3. With this observation it is easily proved that S also preserves the defining relations involving det −1 , and that (2.6) holds for det −1 .
which proves the proposition.
In particular, S is invertible.

The dynamical U (n) quantum group
In this section we prove the existence of a * -operator on F R (GL(n)), such that it becomes a h-Hopf * -algebroid. Equipped with this * -structure we denote the h-Hopf * -algebroid by F R (U (n)).
Lemma 6.1. The * -operator defined on the generators by and extended as C-antilinear algebra anti-homomorphism is well-defined on F R (GL(n)).
Proof. Let I and J be ordered subsets of {1, . . . , n}, such that #I = #J = r. Denote by I c the complement of I in {1, . . . , n}, then we have From this result and Lemma 5.1 it directly follows that * is an involution. The proof of (6.1) is analogous to the corresponding statement (5.6) for the antipode. We prove that * preserves the RLL-relations by using that the antipode does so. By definition of S and * it follows that T a x T c y µ r (R bd xy ).

Using (5.3), * preserves the RLL-relations if
This follows by direct calculations using the explicit expression of R and the fact that sign(x; x) is independent of µ y for all y < x, where the only non-trivial cases are for x = y = b = d, x = b, y = d and x = d, y = b. Using det * = det −1 which follows from (6.1), it directly follows that * preserves the other commutation relations.
From (5.6) and (6.1) it directly follows that

A pairing on the dynamical U (n) quantum group
In this section we discuss pairings for the dynamical GL(n) quantum group and we present a cobraiding on F R (GL(n)). For a pairing for F R (GL(n)) cop and F R (GL(n)) as h-Hopf * -algebroids, we need a second * -operator on F R (GL(n)).

7.1.
Pairing for h-Hopf * -algebroids. We start by recalling the definition of a pairing for h-Hopf * -algebroids.
Definition 7.1. A pairing for h-bialgebroids U and A is a C-bilinear map ·, · : U × A → D h * satisfying for all X ∈ U , a ∈ A. If moreover, U and A are h-Hopf algebroids, then in addition we require If in addition a * -operator is defined on U and A such that then U and A are paired as h-Hopf * -algebroids.
A cobraiding on a h-bialgebroid A is a pairing ·, · : A cop × A → D h * which in addition satisfies as an identity in A and where ∆ A (a) = (a) a (1) ⊗ a (2) , (2) . In [31], Rosengren proved that for a h-bialgebroid constructed by the generalized FRST-construction from an R-matrix, denoted by R, that satisfies the quantum dynamical Yang-Baxter equation (2.1) there exists a natural cobraiding defined on the generators by Note that this is the dynamical analogue of the cobraiding for quantum groups, see e.g. [14,§VIII.6] In [17] we proved the following proposition, which we now extend to the level of h-(co)module algebras. By A lr we denote the h-algebra obtained from a h-algebra A by interchanging the moment maps and with weight spaces (A lr ) αβ = A βα . Proposition 7.3. Let U be a h-algebra and A be h-coalgebroid equipped with a pairing ·, · : U × A → D h * , and let V be a h-space.
(i) Let R : V → V ⊗A be a right corepresentation of the h-coalgebroid A, then π(X)v = (Id ⊗ X, · T β )R(v) for X ∈ U αβ , defines a h-algebra homomorphism π : U → (D h * ,V ) lr , hence π : U lr → D h * ,V defines a dynamical representation of U lr on V .
(ii) Let L : V → A ⊗V be a left corepresentation of the h-coalgebroid A, then π(X)v = (T α X, · ⊗ Id)L(v) for X ∈ U αβ , defines a h-algebra homomorphism π : U opp → (D h * ,V ) lr . In particular, π : (U opp ) lr → D h * ,V defines a dynamical representation of (U opp ) lr on V . Moreover, if U is h-Hopf algebroid , then X → π(S U (X)) defines a dynamical representation of U on V .
Also we compute using the h-module algebra structure of W in the third equation. From (7.7) it follows that Now t j 2 i , t k 2 1 = 0 only if k 2 = i, j 2 = 1 or k 2 = 1, j 2 = i. In the first case, the first leg of the tensor product is equal to 0, so k 2 = 1, j 2 = i. Continuing in this way and recalling that we have pulled the term corresponding to w i to the left, we obtain that there is only a non-zero contribution for j m = i for all m and k 1 = i, k m = m − 1 for 2 ≤ m ≤ i and k m = m for m > i. So we get where the last equality follows from (2.14). So t ii , det = T −1−ω(i) . Note that F R (M (n)) cop can also be seen as a h-bialgebroid constructed from the R-matrixR with matrix elementsR cd ab = R ba dc by the generalized FRST-construction. Following the lines of the proofs of §3 we can prove that V is a right h-comodule algebra for F R (M (n)) cop . By inspection it follows that the matrix elements τ I J of this corepresentation R cop , defined by R cop (v I ) = J v J ⊗ τ I J , are equal to ξ I J . From Proposition 7.5 it follows that π : F R (M (n)) → D h * ,V defined by π(a)v = (Id ⊗ ·, a T β )R cop (v) for a ∈ F R (M n ) αβ and v ∈ V gives V the structure of a h-module algebra for F R (M (n)). Now analogously to the proof of the first part of this lemma we get det, t ii = T −1−ω(i) . Using Lemma 7.6, π(t ij )w 1 · · · w n = 0 if i = j and the explicit expression of det we get π(det)w 1 · · · w n = π(t 11 t 22 · · · t nn )w 1 · · · w n = π(t 11 ) · · · π(t nn )w 1 · · · w n = w 1 · · · w n .

7.3.
Compatible * -structures for the pairing. If we equip F R (GL(n)) cop and F R (GL(n)) with the * -operator defined in Lemma 6.1, they are not paired as h-Hopf * -algebroids. But since the * -operator is not unique it is possible that there exists another * -operator which gives paired h-Hopf * -algebroids.
Proof. The proof follows the lines of the proof of Lemma 6.1. On dynamical quantum minor determinants we have where s I (λ) = q 2(#I/n n k=1 λ k − i∈I λ i ) . This follows using s I\{i} s {i} = s I . From the claim it follows that † is an involution. Indeed, since µ l/r (s I ) det = det µ l/r (s I ) and s {1,...,n} = 1 we have (t † ij ) † = µ l (s i ) µ r (s j ) ξîdet −1 † = det µ l (sî) µ r (s) t ij det −1 µ l (s i ) µ r (s j ) = t ij .
Remark 7.12. Instead of the relation (7.3) we can also require the pairing and * -operator to satisfy a similar relation where * and S are interchanged in the right hand side, see [17]. Also with that relation, the cobraiding (7.6) on the dynamical GL(n) quantum group is not a pairing on the level of h-Hopf algebroids with the same * -operator * on F R (GL(n)) cop and F R (GL(n)).