ERIK KOELINK AND YVETTE VAN NORDEN
Received 4 August 2005; Revised 1 May 2006; Accepted 10 May 2006
We study the dynamical analogue of the matrix algebraM(n), constructed from a dy-namicalR-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 ac-tion of the comultiplicaac-tion 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 dynamicalR-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.
Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.
1. Introduction
Dynamical quantum groups have been introduced recently by Etingof and Varchenko [12], see the review paper by Etingof and Schiffmann [10] for an overview and references to the literature, and related algebraic structures have been studied by Lu [26], Xu [38] in the context of deformations of Poisson groupoids, and by Takeuchi [36]. Brzezi ´nski and Militaru [4] compare the various constructions of [26,36,38]. In this paper we stick to the definition of Etingof and Varchenko [12] with a slight modification as in [20]. In order to keep the paper self-contained as much as possible we recall the definition inSection 2. We also recall the FRST-construction associated to a solution of the dynamicalR-matrix, which gives a wealth of examples, and which we consider explicitly for the trigonometric R-matrix in thegl(n)-case.
It is well-known that quantum groups have a natural link with special functions of basic hypergeometric type, and in [20] it is shown that this remains valid for the sim-plest example of a dynamical quantum group associated to the trigonometricR-matrix for SL(2) and in [22], see also [21], for the ellipticR-matrix for SL(2) giving a dynamical
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International Journal of Mathematics and Mathematical Sciences Volume 2006, Article ID 65279, Pages1–30
quantum group theoretic interpretation of elliptic hypergeometric series. In particular, [20] gives a dynamical quantum group theoretic interpretation of Askey-Wilson andq -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 [24], see also [18,30]. This naturally suggests a link between these two approaches, and the link is established by Stokman [35] using the coboundary element of Babelon et al. [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 prim-itive elements as shown by Rosengren [33]. However, the coboundary element is only known for thesl(2)-case, but there are conjectures about its form for thesl(n) case; see Buffenoir and Roche [5]. The notion of twisted primitive elements of Koornwinder [24], and especially its generalization to coideals, 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, for example, [7,9,25,29,32].
As one of the highlights of the application of Lie theory to special functions we men-tion the group theoretic derivamen-tion of the addimen-tion formula for Jacobi polynomials as obtained by Koornwinder [23] by working on the symmetric spaceU(n)/U(n−1) and establishing the spherical and associated spherical elements in terms of Jacobi polynomi-als; see also Askey [1, Lecture 4] for a nice introduction. In the casen=2 this gives the addition formula for Legendre polynomials. For q-analogues of addition formulas for the Legendre polynomials see the overview [19]. In the quantum group setting, Noumi et al. [31] established the littleq-Jacobi polynomials as spherical functions, and Floris [14] calculated the associated spherical elements in terms of littleq-Jacobi polynomials and derived an addition formula. On the other hand, using the notion of coideals, Di-jkhuizen and Noumi [9] established Askey-Wilson polynomials as spherical functions on a quantum analogue ofU(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 ofU(n)/U(n−1), and if a precise link to special functions can be established. For this we need to study the dynamicalU(n) quantum group more closely, and in a previous paper [21] we have studied general as-pects of dynamical quantum groups for this purpose. In casen=2 the authors in [20] show that the algebraic approach to quantum groups as discussed by Dijkhuizen and Koornwinder [8] is applicable, and we expect this to hold true for generaln. This paper serves as a first step in this specific programme by defining the dynamicalU(n) quantum group and studying some of its elementary properties. In a future paper its corepresenta-tion theory and (associated) spherical funccorepresenta-tions have to be studied.
corepresentation, and we show that the matrix elements, that is, the dynamical quan-tum minor determinants, are equal using an identity for Hall-Littlewood polynomials. In particular, this gives a dynamical quantum determinant. This is done inSection 3. In Section 4, we continue the study of these dynamical quantum minor elements and we discuss the appropriate analogues of the Laplace expansions. InSection 5we 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 anh-Hopf algebroid is very much motivated by the paper by Noumi et al. [31]. We also introduce a ∗-structure, so that we obtain the dynamicalU(n) quantum group inSection 6. Finally, inSection 7we study the natu-ral pairing, as introduced by Rosengren [34], see also [21], for the case of the dynamical GL(n) quantum group and the dynamicalU(n) quantum group.
2. The dynamical analogue of the matrix algebraM(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 theh-bialgebroidᏲR(M(n))
we apply this construction to a solution of the quantum dynamical Yang-Baxter equation (QDYBE).
Lethbe a finite dimensional complex vector space, viewed as a commutative Lie alge-bra, with dual spaceh∗. LetV=α∈h∗Vαbe a diagonalizableh-module. The quantum
dynamical Yang-Baxter equation is given by
R12λ−h(3)R13(λ)R23λ−h(1)=R23(λ)R13λ−h(2)R12(λ). (2.1)
HereR:h∗→End(V⊗V) is a meromorphic function,hindicates the action ofh, and the upper indices are leg-numbering notation for the tensor product. For instance,R12(λ−
h(3)) denotes the operatorR12(λ−h(3))(u⊗v⊗w)=R(λ−μ)(u⊗v)⊗wforw∈V
μ. An
R-matrix is a solution of the QDYBE (2.1) which ish-invariant.
In the example we study, we identifyh∼=h∗∼=Cnand takeV ann-dimensional vector
space with basis{v1,. . .,vn}. TheR-matrixR:h∗→Endh(V⊗V) we consider is given by
R(λ)=q
n
a=1
Eaa⊗Eaa+
a<b
Eaa⊗Ebb+
a>b
gλa−λb
Eaa⊗Ebb+
a=b
h0
λa−λb
Eba⊗Eab,
(2.2)
whereλ=(λ1,. . .,λn),Eab∈End(V) such thatEabvc=δbcvaand the meromorphic
func-tionsh0andgare given by
h0(λ)=
q−1−q
q−2λ−1,
g(λ)=
q−2λ−q−2q−2λ−q2
q−2λ−12 .
Etingof and Varchenko [12] obtain thisR-matrix as the exchange matrix for the vector representation of GL(n), and this R-matrix also fits in Isaev [16] and in particular in the discussion in Hadjiivanov et al. [15]. In the last two references, the corresponding matrixPR(λ), wherePdenotes the flipa⊗b→b⊗a, is considered, which then satisfies a dynamical analogue of the braid relation.
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.
Lethbe a finite dimensional complex vector space, with dual spaceh∗. Denote byMh∗ the field of meromorphic functions onh∗. Forα∈h∗we denote byTα:Mh∗→Mh∗ the automorphism (Tαf)(λ)= f(λ+α) for allλ∈h∗.
Definition 2.1. Anh-algebra is a complex associative algebraᏭwith 1 which is bigraded overh∗,Ꮽ=α,β∈h∗Ꮽαβ, with two algebra embeddingsμl,μr:Mh∗→Ꮽ00(the left and right moment map) such thatμl(f)a=aμl(Tαf),μr(f)a=aμr(Tβf), for all f ∈Mh∗, a∈Ꮽαβ.
A morphism ofh-algebras is an algebra homomorphism which preserves the bigrading and the moment maps.
LetᏭandᏮbe twoh-algebras. The matrix tensor productᏭ⊗Ꮾis theh∗-bigraded vector space with (Ꮽ⊗Ꮾ)αβ=
γ∈h∗(Ꮽαγ⊗Mh∗Ꮾγβ), where⊗Mh∗ denotes the usual ten-sor product modulo the relations
μᏭr (f)a⊗b=a⊗μᏮl (f)b, ∀a∈Ꮽ,b∈Ꮾ, f ∈Mh∗. (2.4)
The multiplication (a⊗b)(c⊗d)=ac⊗bdfora,c∈Ꮽandb,d∈Ꮾand the moment mapsμl(f)=μlᏭ(f)⊗1 andμr(f)=1⊗μᏮr (f) makeᏭ⊗Ꮾinto anh-algebra.
Example 2.2. LetDh∗ be the algebra of difference operators acting onMh∗, consisting of the operatorsifiTβi, with fi∈Mh∗ andβi∈h∗. This is anh-algebra with the bigrading
defined by f T−β∈(Dh∗)ββand both moment maps equal to the natural embedding. For anyh-algebraᏭ, there are canonical isomorphismsᏭ∼=Ꮽ⊗Dh∗∼=Dh∗⊗Ꮽ, de-fined by
x∼=x⊗T−β∼=T−α⊗x, ∀x∈Ꮽαβ. (2.5)
The algebraDh∗plays the role of the unit object in the category ofh-algebras.
Definition 2.3. Anh-bialgebroid is anh-algebraᏭequipped with twoh-algebra
homo-morphismsΔ:Ꮽ→Ꮽ⊗Ꮽ(the comultiplication) andε:Ꮽ→Dh∗ (the counit) such that (Δ⊗Id)◦Δ=(Id⊗Δ)◦Δand (ε⊗Id)◦Δ=Id=(Id⊗ε)◦Δ(under the identifications (2.5)).
Definition 2.4. Anh-Hopf algebroid is anh-bialgebroidᏭequipped with aC-linear map S:Ꮽ→Ꮽ, the antipode, such thatS(μr(f)a)=S(a)μl(f) andS(aμl(f))=μr(f)S(a) for
alla∈Ꮽ, f ∈Mh∗, and
m◦(Id⊗S)◦Δ(a)=μl
ε(a)1, ∀a∈Ꮽ,
m◦(S⊗Id)◦Δ(a)=μrTαε(a)1, ∀a∈Ꮽαβ,
(2.6)
wherem:Ꮽ×Ꮽ→Ꮽdenotes the multiplication andε(a)1 is the result of applying the difference operatorε(a) to the constant function 1∈Mh∗.
If there exists an antipode on anh-bialgebroid, it is unique. Furthermore, the antipode is antimultiplicative, anticomultiplicative, unital, counital and interchanges the moment mapsμl and μr; see [20, Proposition 2.2]. InDefinition 2.4 the mapsm◦(Id⊗S) and
m◦(S⊗Id) are well defined onᏭ⊗Ꮽ; see [21].
Example 2.5. (i) We can equipDh∗with anh-Hopf algebroid structure with comultiplica-tionΔ:Dh∗→Dh∗⊗Dh∗∼=Dh∗ being the canonical isomorphism, counitε:Dh∗ →Dh∗ being the identity, and antipode defined byS(f Tα)=T−α◦f.
(ii) For anh-Hopf algebroidᏭwith invertible antipode, the opposite and cooppo-site are alsoh-Hopf algebroids. The opposite algebraᏭoppis the algebraᏭwith
oppo-site multiplication. Then we equipᏭoppwith anh-Hopf algebroid structure by defining
(Ꮽopp)
αβ=Ꮽ−α,−β,μoppl =μᏭl ,μ
opp
r =μᏭr ,Δopp=ΔᏭ,εopp=SDh∗◦εᏭandSopp=(SᏭ)−1.
The coopposite algebra Ꮽcop has the same algebra structure butμcop
l =μᏭr ,μ
cop
r =μᏭl ,
(Ꮽcop)
αβ=ᏭβαandΔcop=P◦ΔᏭ,εcop=εᏭ,Scop=(SᏭ)−1, wherePis the flip operator.
Letλ→λbe a complex conjugation onh∗, and denotef(λ)= f(λ) for all f ∈Mh∗.
Definition 2.6. Anh-∗-bialgebroidᏭis anh-bialgebroid equipped with a∗-operator, that is, aC-antilinear antimultiplicative involution such thatμl(f)=μl( ¯f) andμr(f)=
μr( ¯f), such that (∗ ⊗ ∗)◦Δ=Δ◦ ∗andε◦ ∗ = ∗Dh∗ ◦ε, where∗ = ∗Dh∗ onDh∗ is defined by (f Tα)∗=T−α¯◦f¯.
Anh-Hopf∗-algebroid is anh-Hopf algebroid that is anh-∗-bialgebroid and has an invertible antipode. Then, see [20],S◦ ∗is an involution.
Until this point we have seen only the exampleDh∗ of anh-bialgebroid. The general-ized FRST-construction provides many examples ofh-bialgebroids fromR-matrices; see [10,11,13,20]. We recall the construction and we apply the construction to theR-matrix in (2.2) to obtain the main object of study for this paper.
LethandMh∗ be as before, letV=α∈h∗Vαbe a finite-dimensional diagonalizable
h-module, and letR:h∗→Endh(V⊗V) be a meromorphic function that commutes
with theh-action onV⊗V. Let{vx}x∈X be a homogeneous basis ofV, whereX is an
index set. WriteRab
xy(λ) for the matrix elements ofR,
R(λ)va⊗vb
=
x,y∈X
and defineω:X→h∗byvx∈Vω(x). LetᏭR be the unital complex associative algebra generated by the elements{Lxy}x,y∈Xtogether with two copies ofMh∗, embedded as sub-algebras. The elements of these two copies will be denoted by f(λ) and f(μ), respectively. The defining relations ofᏭR are f(λ)g(μ)=g(μ)f(λ), f(λ)Lxy=Lxyf(λ+ω(x)), and
f(μ)Lxy=Lxyf(μ+ω(y)) for allf,g∈Mh∗, together with the RLL-relations
x,y∈X
Rxyac(λ)LxbLyd=
x,y∈X
Rbdxy(μ)LcyLax, (2.8)
for alla,b,c,d∈X. The bigrading onᏭRis defined byLxy∈Ꮽω(x),ω(y)and f(λ), f(μ)∈
Ꮽ00. The moment maps defined by μl(f)= f(λ), μr(f)= f(μ) make ᏭR into an
h-algebra. Theh-invariance ofRensures that the bigrading is compatible with the RLL-relations (2.8). Finally the counit and comultiplication defined by
εLab=δabT−ω(a), εf(λ)=εf(μ)= f, ΔLab
=
x∈X
Lax⊗Lxb, Δ
f(λ)=f(λ)⊗1,
Δf(μ)=1⊗f(μ)
(2.9)
equipᏭRwith the structure of anh-bialgebroid; see [11].
Remark 2.7. Hadjiivanov et al. [15] have introduced an analogue of the
FRST-construction involving dynamicalR-matrices. They use the analogue of (2.8) in which theR-matrices may be different, and where it is assumed that theR-matrix on the right-hand side is nondynamical. In case the constantR matrix corresponds to a limit case of the dynamicalR-matrix by taking the limitλ→ ±∞in the dynamical parameter, we see that we can identify the matrix algebra of [15,16] with the limiting case μ→ ±∞. Note that Hadjiivanov et al. [15] do not equip their matrix algebras with a coalgebra type structure.
2.2. The dynamical analogue of the algebraM(n). Now, we apply the generalized FRST-construction to theR-matrix (2.2) in order to define theh-bialgebroidᏲR(M(n)). Let
X= {1,. . .,n}and defineω:X→h∗byi→e
i, whereeiis theith unit vector ofCn. Let
h(λ)=q−h0(λ) whereh0is defined as in (2.3), so
h(λ)=q
q−2λ−q−2
q−2λ−1 . (2.10)
Definition 2.8. Theh-algebraᏲR(M(n)) is the algebra generated by the elementsti j,i,
j∈ {1, 2,. . .,n} together with two copies of Mh∗, denoted by f(λ)= f(λ1,. . .,λn) and f(μ)= f(μ1,. . .,μn), embedded as subalgebras. Then the defining relations are given by
f1(λ)f2(μ)= f2(μ)f1(λ),
with f,f1,f2∈Mh∗, together with the RLL-relations
hμb−μd
tabtad=tadtab, ∀b < d,
hλc−λa
tcbtab=tabtcb, ∀a < c,
tabtcd=tcdtab+hλc−λa−hμb−μd tcbtad, ∀a < c,b < d,
gμb−μd
tabtcd=g
λa−λc
tcdtab+
hμd−μb
−hλa−λc tadtcb, ∀a < c,b < d.
(2.12)
The bigradingᏲR(M(n))=m,p∈NnᏲmp is defined on the generators by f(λ),f(μ)∈
Ᏺ00,ti j∈Ᏺω(i),ω(j)and the moment maps are given byμl(f)=f(λ),μr(f)=f(μ). By
defining the comultiplication Δ:ᏲR(M(n))→ᏲR(M(n))⊗ᏲR(M(n)) and counit ε:
ᏲR(M(n))→Dh∗ on the generators by
Δti j= n
k=1
tik⊗tk j, Δf(λ)= f(λ)⊗1,
Δf(μ)=1⊗f(μ),
(2.13)
andε(ti j)=δi jT−ω(i),ε(f(λ))=ε(f(μ))= f and extended as algebra homomorphisms
we equipᏲR(M(n)) with the structure of anh-bialgebroid.
Remark 2.9. The casen=2 and restricting to functions depending only onλ1−λ2gives
back the case studied in [20]. In case we take a suitable limitμ→ ±∞and restrict to the Cartan subalgebra forsl(n), the algebra also occurs in Hadjiivanov et al. [15], Isaev [16]. However, they do not consider the comultiplication and counit.
As in [31] for the quantum case and in [20] for n=2, we can give a linear basis forᏲR(M(n)). The proof is more involved since we use relations for the functions h,
g.Proposition 2.10is stated for later reference. See also [11, Section 6] for a related result using the Hecke property.
Proposition 2.10. For every n×n-matrix A, denote tA =ta11
11ta1212 ···t1a1nnt21a21···tannnn.
Then{tA:A∈M
n(N)}forms a basis overMh∗⊗Mh∗ for the vector spaceᏲR(M(n)).
Proof. This follows from the diamond lemma; see [3]. First we introduce a total ordering
≺bytA≺tBif i,jai j<
i,jbi j and in case
i,jai j=
i,jbi j we use the lexicographical
ordering on (a11,a12,. . .,a1n,a21,. . .,a2n,a31,. . .,ann).
We have the following reduction system, which is compatible with the introduced total order. Assumei < j,k < l,
tiltik−→hμk−μltiktil, tjktik−→hλj−λi−1tiktjk,
tjltik−→
hλj−λi
−1
+hμl−μk
gλi−λj
−1
tiltjk+g
μk−μl
gλi−λj
−1
tiktjl,
tjktil−→gλi−λj−1tiltjk+
hλj−λi−1−hμk−μlgλi−λj−1
tiktjl.
To simplify the coefficients on the right-hand side we useh(λ)−h(−μ)=h(μ)−h(−λ) andg(μ)−g(λ)=(h(λ)−h(−μ))(h(λ)−h(μ)). If we prove that the reduction system is resolvable, the lemma follows from [3, Theorem 2.1]. There are 24 types of configuration to be checked. The proof is straightforward using
h(−λ)= 1
h(λ+ 1), g(−λ)=g(λ), g(λ)=h(λ)h(−λ),
(2.15)
and the identitiesh(λ)−h(μ)=h(−μ)−h(−λ) and
h(λ)h(λ−1)−h(μ)h(ν) +h(μ−ν)h(ν)−h(μ−ν)h(λ)
+h(ν−μ)h(μ)−h(ν−μ)h(λ)=0, (2.16)
for allλ,μ,ν∈h∗.
3. Exterior corepresentations and dynamical quantum minor determinants
We continue with the study of some elementary corepresentations ofᏲR(M(n))
analo-gous to the action ofM(n) on the exterior algebra ofCn. Using these corepresentations we
find the dynamical determinant inᏲR(M(n)). This is analogous as for the case of
ordi-nary quantum groups; see, for example, Chari and Pressley [6, Chapter 7], and references given there. First we recall the general definition of a corepresentation of anh-bialgebroid on anh-space; see [20]. We introduce the notion ofh-comodule algebras.
Definition 3.1. Anh-space is a vector space overMh∗ which is also a diagonalizable h-module,V=α∈h∗Vα, withMh∗Vα⊆Vαfor allα∈h∗. A morphism ofh-spaces is an h-invariant (i.e., grade preserving)Mh∗-linear map.
In case we want to emphasize the dependence onVwe also write f v=μV(f)v.
We next define the tensor product of anh-bialgebroidᏭand anh-spaceV. PutV⊗Ꮽ=
α,β∈h∗(Vα⊗Mh∗Ꮽαβ) where⊗Mh∗ denotes the usual tensor product modulo the rela-tionsv⊗μl(f)a=f v⊗a. The gradingVα⊗Mh∗Ꮽαβ⊆(V⊗Ꮽ)βfor allαand f(v⊗a)=
v⊗μr(f)amake V⊗Ꮽinto an h-space. AnalogouslyᏭ⊗V =
α,β∈h∗(Ꮽαβ⊗Mh∗ Vβ)
where⊗Mh∗ denotes the usual tensor product modulo the relationsμr(f)a⊗v=a⊗f v.
The gradingᏭαβ⊗Mh∗Vβ⊆(Ꮽ⊗V)αand f(a⊗v)=μl(f)a⊗v,a∈Ꮽ,v∈V,f ∈Mh∗, makeᏭ⊗V into anh-space.
Definition 3.2. A right corepresentation of anh-bialgebroidᏭon anh-space V is an
h-space morphism ρ:V →V⊗Ꮽsuch that (Id⊗Δ)◦ρ=(ρ⊗Id)◦ρ, (Id⊗ε)◦ρ=Id. The first equality is in the sense of the natural isomorphism (V⊗Ꮽ)⊗Ꮽ∼=V⊗(Ꮽ⊗Ꮽ)
and in the second identity, use the identificationV∼=V⊗Dh∗ defined byv⊗f T−α∼= f v, f ∈Mh∗, for allv∈Vα.
Definition 3.3. LetᏭbe anh-bialgebroid andV anh-space. ThenV is a right (left)h -comodule algebra forᏭif there exists a right (left) corepresentationR:V →V⊗Ꮽ(L: V→Ꮽ⊗V) such that
(i)V is an associative algebra such thatμV(f)vw=vμV(Tαf)w forv∈Vα,w∈V,
andVαVβ⊂Vα+β,
(ii)R(L) is an algebra homomorphism.
If, moreover,Vis a unital algebra, we requireR(L) to be unital.
Remark 3.4. The algebra structure of V⊗Ꮽ is given by (v⊗a)(w⊗b)=vw⊗ab for v,w∈V anda,b∈Ꮽ. Forv∈Vα,w∈Vγ anda∈Ꮽαβ,b∈Ꮽγδ, we have (v⊗a)(w⊗
b)=vw⊗ab∈Vα+γ⊗Ꮽα+γ,β+δ using (i) which implies (V⊗Ꮽ)β(V⊗Ꮽ)δ⊆(V⊗Ꮽ)β+δ.
Forv∈Vα,
μV⊗A(f)R(v)=1⊗μr(f)
R(v)=R(v)1⊗μr
Tαf
=R(v)μV⊗ATαf
. (3.1)
SoRpreserves the relation in (i). Recall that by theMh∗-linearity of a corepresentation we haveR(μV(f)v)=μV⊗A(f)R(v)=(1⊗μr(f))R(v).
Now we define the h-space W on which we construct a right corepresentation of ᏲR(M(n)).Wcan be seen as the dynamical analogue of the exterior algebra
represen-tation.
Definition 3.5. LetWbe the unital associative algebra generated by the elementswi,i∈
{1, 2,. . .,n}and a copy ofMh∗ embedded as a subalgebra, its elements denoted by f(λ), subject to the relations
w2i =0, ∀i, wjwi= −h
λj−λi
wiwj, ∀i < j, (3.2)
withhdefined by (2.10) and f(λ)wi=wif(λ+ω(i)) for all f ∈Mh∗.
For an ordered subsetI= {i1,. . .,ir}, 1≤i1<···< ir≤n, of{1,. . .,n}we use the
con-ventionwI=wi1···wir, unless mentioned otherwise. Moreover,∅is an ordered subset
andw∅=1 corresponding to the caser=0. The following lemma is easily proved. Lemma 3.6. dimMh∗W=2nand a basis forWis given by{wI:I= {i1,. . .,ir}, i1<···<
ir,r=1,. . .,n}.Wis anh-space withμW(f)=f(λ) andwI∈Wω(I)forwIa basis element
withω(I)=r j=1ω(ij).
DefineWr=span
Mh∗{wI: #I=r}. ThenW=
n
r=0WrandWrWs⊂Wr+s, with the
convention thatWr= {0}ifr > n.
Proposition 3.7. DefineR(1)=1⊗1,R(wi)=
n
j=1wj⊗tji. ThenRextends uniquely to
R:W→W⊗ᏲR(M(n)) such thatWis a righth-comodule algebra forᏲR(M(n)).
Proof. It is clear thatWsatisfies the conditions ofDefinition 3.3(i). To see thatRcan be extended uniquely to an algebra homomorphism we need to verify
Rwi
Rwi
=0, ∀i, Rwj
Rwi
= −Rhλj−λi
Rwi
Rwj
and R(f(λ))R(wi)=R(wi)R(f(λ+ω(i))) for all f ∈Mh∗. By definition of R and the defining relations (3.2) ofWwe get
Rwi
Rwi
=
j,k
wjwk⊗tjitki=
k> j
wjwk⊗tjitki−h
λk−λj
wjwk⊗tkitji (3.4a)
=
k> j
wjwk⊗
tjitki−h
λk−λj
tkitji
=0, (3.4b)
where we use the second relation of (2.12) in the last equality. Let us emphasize that the functionhshould be interpreted in (3.4a) asμW(λ→h(λk−λj)) and in (3.4b) asμl(λ→
h(λk−λj)). Similarly we obtain that the relationR(wj)R(wi)= −R(h(λj−λi))R(wi)R(wj)
fori < jis equivalent to
l>k
wkwl⊗tk jtli−hλl−λktl jtki+hμj−μitkitl j−hμj−μihλl−λktlitk j=0.
(3.5)
UsingLemma 3.6it remains to prove that
tk jtli−h
λl−λk
tl jtki+h
μj−μi
tkitl j−h
μj−μi
hλl−λk
tlitk j=0, (3.6)
fori < j,k < l. To show this we multiply this equation byh(λl−λk)−h(μi−μj) and
elimi-nate the productstk jtliandtlitk jusing the third and fourth relations in (2.12), respectively.
Usingh(λ)−h(−μ)=h(μ)−h(−λ) for allλ,μwe obtain that the relation (3.6) holds by (2.15). Using the definition ofRandRemark 3.4the last relation follows analogously.
By the definition of the comultiplication and the counit on the generators ofᏲR(M(n))
of Definition 2.8 it immediately follows that (Id⊗Δ)◦R(wi)=(R⊗Id)◦R(wi) and
(Id⊗ε)◦R(wi)=wi. SinceR,Δ, andεareh-algebra homomorphisms, so are (Id⊗Δ)◦R,
(R⊗Id)◦R, and (Id⊗ε)◦R. So the equalities (Id⊗Δ)◦R=(R⊗Id)◦Rand (Id⊗ε)◦R=
Id hold on the generators and hence on all ofW.Ris a corepresentation ofᏲR(M(n)) on
WandWis anh-comodule algebra forᏲR(M(n)).
ForIandJordered subsets with #I=#Jwe define the elementsξI
J as the corresponding
matrix elements;
RwJ=
#I=#J
wI⊗ξJI. (3.7)
We use the convention thatξJI=0 for allI,J such that #I=#J. In the remainder of the
paper use the convention that a summation over subsets such as#I=r is a summation
over all ordered subsetsIsuch that #I=r. Corollary 3.8. (i)Δ(ξI
J)=
#K=#IξKI ⊗ξJKandε(ξJI)=δIJT−ω(I)for allI,Jwith #I=#J.
(ii)R(Wr)⊂Wr⊗Ᏺ
R(M(n)).
We call the matrix elements ξJI the dynamical quantum minor determinants of ᏲR(M(n)) with respect to the subsetsI andJ. The elementξ{{1,1,......,,nn}} is called the
This right corepresentation has a left analogue, a left h-comodule algebra V for ᏲR(M(n)). The proofs are analogous to the ones for the righth-comodule algebraW,
and are skipped.
Definition 3.9. LetV be the unital associative algebra generated by the elementsvi,i∈
{1,. . .,n} and a copy ofMh∗ embedded as a subalgebra, its elements denoted by f(λ), subject to the relations
v2
i =0, ∀i, vivj= −h
λi−λj
vjvi, ∀i < j, (3.8)
and f(λ)vi=vif(λ+ω(i)) for all f ∈Mh∗.
For an ordered subsetI= {i1,. . .,ir}with 1≤i1<···< ir≤n we denote byvI the
ordered elementvI=vir···vi1∈V. Let us emphasize that an elementvI∈Vhas reversed
order compared towI∈Wby notational convention.
Lemma 3.10. dimMh∗V=2nand a basis forV is given by{vI:I= {i1,. . .,ir},i1<···<
ir,r=1,. . .,n}.V is anh-space withμV(f)=f(λ) andvI∈Vω(I)forvIa basis element.
DefineVr=span
Mh∗{vI: #I=r}. ThenV=
n
r=0VrandVrVs⊂Vr+s, with the
con-vention thatVr= {0}ifr > n.
Proposition 3.11. DefineL(1)=1⊗1,L(vi)=nj=1ti j⊗vj. ThenLextends uniquely to
L:V→ᏲR(M(n))⊗Vsuch thatVis a lefth-comodule algebra forᏲR(M(n)).
For ordered subsetsI,Jwith #I=#Jwe define the elementsηI
J by
LvI
=
#J=#I
ηI
J⊗vJ (3.9)
andηJI=0 for #I=#J. We denote the corresponding determinant bydet=η{{1,1,......,,nn}}.
Corollary 3.12. (i)Δ(ηIJ)=
#K=#IηKI ⊗ηJKandε(ηIJ)=δIJT−ω(I)forI,Jwith #I=#J.
(ii)L(Vr)⊂Ᏺ
R(M(n))⊗Vr.
We call the matrix elements ηI
J the dynamical quantum minor determinants of
ᏲR(M(n)) with respect to the subsetsI andJ. InTheorem 3.17we prove that the
dy-namical quantum minor determinants related to the right and left corepresentations are equal, so we can speak of the dynamical quantum minor determinants ofᏲR(M(n)),
without mentioning right or left. First we compute an explicit expression of the dynami-cal quantum minor determinants which we use in the proof.
For any permutationσ∈Sr, 1≤r≤n, and any ordered subsetI= {i1,. . .,ir}, we
de-fine the generalized sign functionS(σ,I)∈Mh∗by
S(σ,I)(λ)=
{k<l:σ(k)>σ(l)}
−hλiσ(k)−λiσ(l)
=(−q)l(σ) {k<l:σ(k)>σ(l)}
q−2λiσ(k)−q−2q−2λiσ(l)
q−2λiσ(k)−q−2λiσ(l) ,
(3.10)
Lemma 3.13. For any permutationσ∈Srthe following relation inWholds:
wiσ(1)···wiσ(r)=μW
S(σ,I)wI, (3.11)
whereI= {i1,. . .,ir}is ordered.
Proof. We prove by induction onr, forr=2 andσ=Id it is trivial. Ifσ=(12) it is just (3.2) forj=iσ(1),i=iσ(2). Denote byIthe ordered subset ofIdefined byI\ {iσ(1)}, then
wiσ(1)···wiσ(r+1)=wiσ(1)
2≤k<l≤r+1
σ(k)>σ(l)
−hλiσ(k)−λiσ(l)
wI=
2≤k<l≤r+1
σ(k)>σ(l)
−hλiσ(k)−λiσ(l)
2≤l≤r+1
σ(1)>σ(l)
−hλiσ(1)−λiσ(l)
wI=μW
S(σ,I)wI,
(3.12)
sincewiσ(1)commutes with all functions inMh∗ which are independent ofλiσ(1).
UsingLemma 3.13we calculate the action of the corepresentationRonwj1···wjrfor
an arbitrary unordered set{j1,. . .,jr}. Then
Rwj1···wjr
=Rwj1
···Rwjr
=
n
k1=1 ···
n
kr=1
wk1···wkr⊗tk1j1···tkrjr, (3.13)
and there is only a nonzero contribution in the right-hand side of (3.13) if allki=kj
fori= j. LetI= {i1,. . .,ir}be ordered, then we see that the contribution on the
right-hand side of (3.13) containing the basis elementwIin the first leg of the tensor product is
given for those terms for which{i1,. . .,ir} = {k1,. . .,kr}as unordered sets. So there exists
for each nonzero term in (3.13) contributing to the term containingwI in the first leg
of the tensor product precisely one permutationσ∈Srsuch thatkp=iσ(p). So the term
containingwIin the first leg of the tensor product equals
σ∈Sr
wiσ(1)···wiσ(r)⊗tiσ(1)j1···tiσ(r)jr=
σ∈Sr
μW
S(σ,I)wI⊗tiσ(1)j1···tiσ(r)jr
=
σ∈Sr
wI⊗μl
S(σ,I)tiσ(1)j1···tiσ(r)jr,
(3.14)
byLemma 3.13andRemark 3.4.
Proposition 3.14. Let J be ordered withr=#J, thenR(wJ)=
#I=#JwI⊗ξJI with the
dynamical quantum minor determinants given by
ξI J =μr
S(ρ,J)−1
σ∈Sr
μl
S(σ,I)tiσ(1)jρ(1)···tiσ(r)jρ(r), (3.15)
Proof. ByLemma 3.13and the discussion preceding this proposition we obtain
#I=#J
wI⊗ξJI=R
wJ
=1⊗μr
S(ρ,J)−1Rwjρ(1)···wjρ(r)
=1⊗μrS(ρ,J)−1
#I=#J
wI⊗μlS(σ,I)tiσ(1)jρ(1)···tiσ(r)jρ(r).
(3.16)
So, the proposition follows fromLemma 3.6.
Corollary 3.15. PutS(σ)=S(σ,{1,. . .,n}) forσ∈Sn, then for anyρ∈Sn,
det=μr
S(ρ)−1
σ∈Sn
μl
S(σ)tσ(1)ρ(1)···tσ(n)ρ(n). (3.17)
Analogously we obtain an explicit formula for the matrix elementsηIJofL. We need to
define another generalized sign functionSdepending on an ordered subsetI, #I=r, and a permutationσ∈Sr;
S(σ,I)(λ) :=
{k<l:σ(k)>σ(l)}
−hλiσ(l)−λiσ(k)
= 1
S(σ,I)(λ+ 1), (3.18)
where we useh(−λ)=1/h(λ+ 1) for the last equality. Analogous toLemma 3.13, we have for any permutationσ∈Srthe following relation inV:
viσ(r)···viσ(1)=μVS(σ,I)
vI, (3.19)
whereI= {i1,. . .,ir}is an ordered subset andvI=vir···vi1. We get the analogous
state-ment ofProposition 3.14.
Proposition 3.16. Let I= {i1,. . .,ir} be an ordered subset, thenL(vI)=
#J=#IηIJ⊗vJ
with the dynamical quantum minor determinants given by, for anyρ∈Sr,
ηIJ=μlS(ρ,I)−1 σ∈Sr
μrS(σ,J)tiρ(r)jσ(r)···tiρ(1)jσ(1). (3.20)
We now relate the two sets of dynamical quantum minor determinants. For this we need the following identity:
σ∈Sr
i< j
xσ(i)−txσ(j)
xσ(i)−xσ(j) =
r
i=1
1−ti
1−t, (3.21)
forrindeterminatesx1,. . .,xr. This identity can be found in Macdonald [27, Section III.1,
equation (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 (3.15) and (3.20), which give the possibil-ity to write a suitable multiple ofξJI as a double sum overSr, which, by interchanging
summations, gives a multiple ofηIJ. The multiples turn out to be equal. The details are as follows.
First we rewriteηI
J. Define the longest elementσ0∈Sr byσ0=(1r r−21······r1). By
substi-tutingρ→ρσ0andσ→σσ0in (3.20) we get
ηIJ=
m<p ρ(m)<ρ(p)
−hλiρ(m)−λiρ(p)
−1
σ∈Sr
k<l σ(k)<σ(l)
−hμjσ(k)−μjσ(l)
tiρ(1)jσ(1)···tiρ(r)jσ(r), (3.22)
for anyρ∈Sr. Using this expression forηIJ and (3.15) we compute
ρ∈Sr
k<l
−hμjρ(k)−μjρ(l)
ξI J
=
ρ∈Sr
σ∈Sr
k<l ρ(k)<ρ(l)
−hμjρ(k)−μjρ(l) k<l σ(k)>σ(l)
−hλiσ(k)−λiσ(l)
tiσ(1)jρ(1)···tiσ(r)jρ(r)
=
σ∈Sr
ρ∈Sr
k<l
−hλiσ(k)−λiσ(l) k<l σ(k)<σ(l)
−hλiσ(k)−λiσ(l)
−1
×
k<l ρ(k)<ρ(l)
−hμjρ(k)−μjρ(l)
tiσ(1)jρ(1)···tiσ(r)jρ(r)=
σ∈Sr
k<l
−hλiσ(k)−λiσ(l)
ηI J.
(3.23)
So it suffices to prove thatA(I)(λ) :=ρ∈Sr
k<l−h(λiρ(k)−λiρ(l)) is independent ofλand
I:
A(I)(λ)=
ρ∈Sr
k<l
(−q)q
−2λiρ(k)−q−2q−2λiρ(l)
q−2λiρ(k)−q−2λiρ(l) =(−q)
(1/2)r(r−1)
r
k=1
1−q−2k
1−q−2 =0, (3.24)
using the explicit expression (2.10) forhand (3.21).
Corollary 3.18. det=det.
Remark 3.19. (i) The dynamical quantum minor determinantξJI belongs to the weight
spaceᏲR(M(n))ω(I),ω(J), whereω(I)= r
k=1ω(ik)=
r
k=1eik,I= {i1,. . .,ir}.
(ii) FromTheorem 3.17we obtain relations inᏲR(M(n)). Forr=2 we get quadratic
relations for the generatorsti j, forρ=Id in the expressions of ξJI andηIJ in (3.15) and
Remark 3.20. Hadjiivanov et al. [15] introduce a dynamical determinant in the dynamical analogue ofM(n) they consider; see also Remarks2.7,2.9. In particular, they show that their dynamical quantum determinant is central; (cf.Lemma 5.1). They work with the Hecke property of the matrixPR(λ), and use theq-antisymmetrisers in the Hecke algebra of typeAfollowing the ideas of Gurevich for the quantum SL(n)-group; see [6, Section 7.2] for more information.
4. 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.
ForI1,I2 disjoint ordered subsets of{1,. . .,n}, denote by sign(I1;I2) the element of
Mh∗defined by
signI1;I2
(λ)=
k>m k∈I1,m∈I2
−hλk−λm. (4.1)
ThenwI1wI2=μW(sign(I1;I2))wIifI1∩I2=∅andI1∪I2=I. IfI1∩I2= ∅, thenwI1wI2=
0 and in this case we define sign(I1;I2)(λ)=0. ForI1∩I2= ∅andI=I1∪I2as ordered
subset we have sign(I1;I2)=S(σ,I) whereσis the permutation which mapsI1∪I2to the
ordered subsetI.
Proposition 4.1 (Laplace expansions). LetI,J1,J2be subsets of{1,. . .,n}. IfJ=J1∪J2,
#J=#I,
μrsignJ1;J2
ξJI=
I1∪I2=I
μlsignI1;I2
ξJ1I1ξJ2I2,
μl
T−ωJ1signJ2;J1 −1
ξIJ=
I1∪I2=I
μr
T−ωI1signI2;I1 −1
ξI1J1ξI2J2,
(4.2)
where the summation runs over all partitionsI1∪I2=IofIsuch that #I1=#J1, #I2=#J2. Remark 4.2. (i) Note that the left-hand sides of the expressions in (4.2) are zero ifJ1and
J2are not disjoint.
(ii) The second relation of (4.2) can be rewritten as
ξIJ=
I1∪I2=I
ξI1J1μl
signJ2;J1
μr
signI2;I1
ξI2J2. (4.3)
Proof ofProposition 4.1. We have
RwJ1
RwJ2
=
I1∩I2=∅
wI1wI2⊗ξJ1I1ξJ2I2
=
I1∩I2=∅ μW
signI1;I2
wI⊗ξJ1I1ξJ2I2
=
I1∩I2=∅
wI⊗μlsignI1;I2ξJ1I1ξJ2I2.
Also, ifJ1∩J2= ∅, thenR(wJ1wJ2)=R(0)=0 by (3.2) which proves the first relation of
(4.2) usingLemma 3.6in the case thatJ1 andJ2are not disjoint. IfJ1∩J2= ∅, then we
also have
RwJ1RwJ2=RwJ1wJ2=1⊗μrsignJ1;J2
RwJ=
#I=#J
wI⊗μrsignJ1;J2
ξI J.
(4.5)
The second relation of (4.2) is proved analogously, usingLinstead ofRandTheorem 3.17.
In the special case #I=#J=nand eitherJ1orJ2contains one element, we get the
fol-lowing 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 deter-minant of a matrix.
Corollary 4.3. For alliandjwith 1≤i, j≤n,
δi jdet= n
k=1
sign({k};k)(λ) sign({i};ı)(μ)tk jξ
k
ı, δi jdet=
n
k=1
tjk sign(ı;{i})(λ)
sign(k;{k})(μ)ξ
ı
k,
δi jdet= n
k=1
sign(k;{k})(λ) sign(ı;{i})(μ)ξ
k
ıtk j, δi jdet= n
k=1
ξkı sign({i};ı)(λ) sign({k};k)(μ)tjk,
(4.6)
with the notationı= {1,. . .,i−1,i+ 1,. . .,n}.
5. The dynamical GL(n) quantum group
In this section we extend ᏲR(M(n)) by adjoining an inverse of the determinant. The
resultingh-bialgebroidᏲR(GL(n)) is equipped with an antipode, so it is anh-Hopf
alge-broid.
Lemma 5.1. InᏲR(M(n)), the determinant element commutes with all quantum minor
determinantsξJI, forI,Jsubsets of{1,. . .,n}. In particular, det commutes with all generators
ti j. Moreover,Δ(det)=det⊗det andε(det)=T−1, with 1=(1,. . ., 1)∈h∗.
Proof. Denote byTthen×n-matrix with elementsti j, whereiindicates the row index.
Using the notation
Ti
j=
μl
sign(ı;{i}) μr
sign(j;{j})ξ
ı
j, (5.1)
denote byTthen×n-matrix with elementsTi
jwhereiindicates the column index. Then
the third relation ofCorollary 4.3impliesTT =detIasn2identities inᏲ
R(M(n)), where
Iis then×n-identity matrix. So detT=TTT =Tdet which implies that det commutes with all generatorsti j. Since det∈ᏲR(M(n))1,1, we see that det commutes with all
So the determinant element commutes with all generators ti j, but since det∈
ᏲR(M(n))1,1, the element det is not central. However, the setS= {detk}k≥1 satisfies the
Ore condition, and this implies that we can localize at det; see [28]. We adjoinᏲR(M(n))
with the formal inverse det−1, adding the relations det det−1=1=det−1det, ti jdet−1=
det−1ti jandf(λ) det−1=det−1f(λ−1),f(μ) det−1=det−1f(μ−1). We denote the resulting
algebra byᏲR(GL(n)) and equip it with a bigradingᏲR(GL(n))=
m,p∈ZnᏲR(GL(n))mp
by det−1∈(ᏲR(GL(n)))−1,−1.Lemma 5.1implies that det−1commutes with all
dynam-ical quantum minor determinantsξJI. By extending the comultiplication and counit of Definition 2.8byΔ(det−1)=det−1⊗det−1,ε(det−1)=T1,ᏲR(GL(n)) it is easily checked
thatᏲR(GL(n)) is anh-bialgebroid.
Proposition 5.2. Theh-bialgebroidᏲR(GL(n)) is anh-Hopf algebroid with the antipode
Sdefined on the generators byS(det−1)=det,S(μr(f))=μl(f),S(μl(f))=μr(f) for all
f ∈Mh∗and
Sti j
=det−1μl
sign(j;{j}) μr
sign(ı;{i})ξ
j
ı, (5.2)
and extended as an algebra antihomomorphism.
Proof. By [20, Proposition 2.2] it suffices to check thatSis well defined and that (2.6) holds on the generators. It is straightforward to check thatSpreserves the relations (2.11). To see thatSpreserves the RLL-relations, we apply the antipode to the RLL-relations (2.8). Using (5.1) this gives
x,y
det−2Td yTxbR
xy ac(μ)=
x,y
det−2Tx aT
y
cRbdxy(λ), (5.3)
which is equivalent to
x,y
Rxyacμ+ω(x) +ω(y)TdyTxbdet−
2
=
x,y
det−2Tx aT
y
cRbdxy(λ). (5.4)
We have to prove that (5.4) holds in ᏲR(GL(n)). To show this, we multiply the
RLL-relations (2.8) byTdkTbl from the right and byTajTicfrom the left and sum over alla,b,c, anddwe get, usingCorollary 4.3,
a,c
TajTicRlkac(λ)det2=
b,d
det2Rbdji
μ+ω(i) +ω(j)TdkTbl. (5.5)
Multiplying this equation from the left and from the right by det−2 gives (5.4) by the h-invariance of theR-matrix, soSpreserves the RLL-relations.
From the proof ofLemma 5.1it follows thatS(T)T=TS(T)=I, whereTis defined as in the proof ofLemma 5.1, so (2.6) holds for all generatorsti j. The proof of [20, Prop.
holds for det. ByLemma 5.1we findS(det) det=1=detS(det), so thatS(det)=det−1. An independent proof of this statement is given inProposition 5.3. With this observation it is easily proved thatSalso preserves the defining relations involving det−1, and that
(2.6) holds for det−1.
The relationS(det)=det−1is the special caseI=J= {1,. . .,n}of the following propo-sition.
Proposition 5.3. ForIandJordered subsets with #I=#J,
SξJI
=det−1μl
signJc;J
μr
signIc;Iξ Jc
Ic, (5.6)
withIcthe complement ofIin{1,. . .,n}.
Proof. We prove this formula by induction on ther:=#Iusing the Laplace expansions of Proposition 4.1. Another proof uses (2.6) combined with the Laplace expansions. We use a similar induction step in the proof ofLemma 6.1.
Forr=1 this is just the definition of the antipode on a generatorti j. For the
induc-tion step we use the Laplace expansions ofProposition 4.1several times. Let j∈J, then applying the Laplace expansion twice
SξJI
=S
i∈I
μl
sign({i};I) μr
sign({j};J)ξ {i} {j}ξI
J
=
i∈I
SξJI
Sξij
μr
sign({i};I) μl
sign({j};J)
=
i∈I
det−2μl
sign(Jc;J) μr
sign(Ic;I)ξ
Jc Ic
μl
sign(j;{j}) μr
sign(ı;{i})ξ
j
ı
μr
sign({i};I) μl
sign({j};J)
=
i∈I
det−2μl
signJc;J μr
signIc;Iξ
Jc Ic
μl
sign(j;{j}) μr
sign(ı;{i})ξ
j
ı
=
i∈I
k∈Jc K=Jc\{k}
det−2μl
signJc;J μr
signIc;I μl
sign(K;{k}) μr
signIc;{i}ξ K Icξ{{ik}}
μl
sign(j;{j}) μr
sign(ı;{i})ξ
j
ı,
(5.7)
whereJ=J\ {j}andI=I\ {i}(soIdepends on the summation index) as ordered subsets. In this computation we use
ξIJccξıj
μrsign({i};I)
μl
sign({j};J)=
μrsign({i};I)
μl
sign({j};J)ξ
Jc
Icξıj (5.8)
sinceξIJccξıjhas weight (1 +ω(Jc), 1 +ω(Ic)) and sign({a};B) sign(A;B)=sign({a} ∪A;B)
for all subsetsA,Band all elementsa∈A. Since k∈Jc
K=Jc\{k}μl(sign(K;{k}))ξ K
alli∈Icand sign(ı;{i})=sign(I;{i}) sign(Ic;{i}) we obtain, using the Laplace
expan-sion once more for the summation overiwhere the only nonzero term is fork=j,
SξI J
=
k∈Jc K=Jc\{k}
det−2μl
signJc;J μrsignIc;Iμl
sign(K;{k})ξK Ic
n
i=1
ξ{{ik}}μl
sign(j;{j}) μrsign(ı;{i})ξ
j
ı
=det−2μl
signJc;J μrsignIc;Iμl
signJc;{j}ξJc
Icdet=det−1
μl
signJc;J
μrsignIc;Iξ Jc Ic,
(5.9)
which proves the proposition.
Corollary 5.4.
S2ξJI
=
m∈I,k∈Ich
λm−λk
m∈J,k∈Jch
μm−μk
ξJI. (5.10)
In particular,Sis invertible.
6. The dynamicalU(n) quantum group
In this section we prove the existence of a∗-operator onᏲR(GL(n)), such that it
be-comes anh-Hopf ∗-algebroid. Equipped with this∗-structure we denote the h-Hopf
∗-algebroid byᏲR(U(n)).
Lemma 6.1. The∗-operator defined on the generators by
t∗i j=ξı
jdet−1, μl(f)∗=μl(f), μr(f)=μr(f),
det−1∗=det, (6.1)
and extended asC-antilinear algebra anti-homomorphism is well defined onᏲR(GL(n)).
Proof. LetIandJbe ordered subsets of{1,. . .,n}, such that #I=#J=r. Denote byIcthe
complement ofIin{1,. . .,n}, then we have
ξI
J
∗=ξIc
Jcdet−1. (6.2)
From this result andLemma 5.1it directly follows that∗is an involution. The proof of (6.2) 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 ofSand∗it follows that
μr
sign(k;{k})Stk j
=μl
Applying∗to the RLL-relations (2.8) we get
n
x,y=1
μr
sign(d;{d}) μl
sign(y;{y})T
y d
μr
sign(b;{b}) μl
sign(x;{x})T
x bμl
Rxyac
=
n
x,y=1
μrsign(x;{x})
μl
sign(a;{a})T
a x
μrsign(y;{y})
μl
sign(c;{c})T
c yμr
Rbd xy , (6.4)
which is equivalent to
n
x,y=1
μl
sign(a;{a}) μl
sign(y;{y}) μl
Tω(a)sign(c;{c})
μl
Tω(y)sign(x;{x})
TdyTbxμl
Rxyac
=
n
x,y=1
μr
sign(x;{x}) μr
sign(d;{d}) μr
Tω(x)sign(y;{y})
μr
Tω(d)sign(b;{b}) Ta
xTcyμr
Rbd xy . (6.5)
Using (5.3),∗preserves the RLL-relations if
Rdbyx(μ)=Rbd xy(μ)
sign(x;{x})(μ−ω(x)−ω(y) sign(d;{d})(μ−ω(x)−ω(y)
sign(y;{y})(μ−ω(y)
sign(b;{b})(μ−ω(b). (6.6)
This follows by direct calculations using the explicit expression ofR and the fact that sign(x;x) is independent ofμyfor all y < x, where the only nontrivial cases are forx=
y=b=d,x=b,y=dandx=d,y=b. Using det∗=det−1which follows from (6.2), it directly follows that∗preserves the other commutation relations. Proposition 6.2. Denote ᏲR(GL(n)) equipped with the ∗-operator of Lemma 6.1 by
ᏲR(U(n)), thenᏲR(U(n)) is anh-Hopf∗-algebroid.
Proof. From the definition of∗andCorollary 3.8it follows that (∗ ⊗ ∗)Δ(ti j)=Δ(ti j∗)
and (ε◦ ∗)(ti j)=(∗Dh∗◦ε)(ti j),
(∗ ⊗ ∗)Δdet−1=det⊗det=Δdet−1∗,
(ε◦ ∗)det−1=T−1=
T1
∗=εdet−1∗. (6.7)
So the relations (∗ ⊗ ∗)◦Δ=Δ◦ ∗ and ε◦ ∗ = ∗Dh∗◦ε hold on the generators of
ᏲR(GL(n)) and hence on all ofᏲR(GL(n)).
From (5.6) and (6.2) it directly follows that
SξI J
∗=ξJ I
μlsignJc;J
μr
signIc;I, S
ξI
J
∗= μlsignJ;Jc
μr
signI;Icξ J
I, (6.8)
which gives an indication for the unitarisability of the corepresentations R and L of ᏲR(GL(n)) defined in Propositions3.7and3.11; for the definition of unitarisability see
Proposition 6.3. The corepresentations R and L are unitarisable corepresentations of
ᏲR(U(n)).
Proof. We have to define a form·,·:W×W→Mh∗ and check thatR(x),R(y) = μr(x,y1) for allx,y∈W; see [21, Section 5]. It is sufficient to do this for basis
el-ements{wI}of W. Define wI,wJ(λ)=δIJsign(Ic;I)(λ−ω(I))∈Mh∗, sowI,wJD= δIJsign(Ic;I)∈Dh∗. Then
RwI
,RwJ
=
#K=#J
wK⊗ξJK,
#M=#J
wM⊗ξJM
=
K,M
wK,wM
D⊗
ξM
J
∗ξK J
=
K
μlsignKc;K μr
signJc;J
μlsignKc;KS
ξKJξIK=μrsignJc;JδIJ
=μr
wI,wJ
D1