Dynamical quantum groups: Duality and special functions

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Dynamical Quantum Groups

Duality and Special Functions

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Dynamical Quantum Groups

Duality and Special Functions

Proefschrift

ter verkrijging van de graad \'an doctor aan de Technische Universiteit Delft,

op gezag van de Rector Magniflcus prof.dr.ir J.T. Fokkema voorzitter van het College voor Promoties,

in het openbaar te verdedigen op dinsdag 13 september 2005 om 13.00 uur

door Yvette VAN NORDEN wiskundig ingenieur geboren te Rotterdam.

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Dit proefschrift is goedgekeurd door de promotor: Prof.dr. Ph.P.J.E. Clément

Samenstelling promotiecommissie: Rector Magnificus

Prof.dr. Ph.P.J.E. Clément Prof.dr.ir. A.W. Heemink Prof.dr. T.H. Koornwinder Prof.dr. H.G. Meijer Prof.dr. H. Rosengren Prof.dr. A.J. van Zanten dr. J.V. Stokman

voorzitter

Technische Universiteit Delft, promotor Technische Universiteit Delft

Universiteit van Amsterdam Technische Universiteit Delft

Chalmers University of Technology and Göteborg University, Zweden

Universiteit Maastricht Universiteit van Amsterdam

Het onderzoek beschreven in dit proefschrift is mede gefinancierd door de Neder-landse Organisatie voor Wetenschappelijk Onderzoek (NWO), onder projectnnmmer 613.006.572.

Het Stieltjes Instituut heeft bijgedragen in de drukkosten van het proefschrift.

T H O M A S S T I E L T J E S I N S T I T U T E FOR M A T H E M A T I C S

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Preface

Some of the results presented in this thesis have been published in [34], [32] and [33]. Chapter 2 is mainly based on [32]. The main results of Chapter 3 have been obtained originally in a different way, which is the content of [34]. The proofs in Chapter 3 are different from those in [34] and are now in line with Chapter 2. The results of Chapter 4 will appear in [33].

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het is een manier om wijs te zijn. C o l e t t e

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Bibliography 125

Summary 131

Samenvatting I33

Dankwoord I35

Curriculum Vitae I37

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Chapter 1 Introduction

In this thesis we study dynamical quantum groups and their relation to special func-tions, in particular elliptic hypergeometric series. Although dynamical quantum groups are no groups, they have many features in common with groups. In particular, there are well-defined analogues of unitary group representations. In Chapter 2 we discuss pairings and actions for dynamical quantum groups, these actions can be seen as ana-logues of the left and right action of the universal enveloping algebra of a Lie algebra on the algebra of functions on the corresponding Lie group by left and right invariant differential operators. There is a well-known relation between the representation the-ory of specific groups, such as groups of matrices, and special functions. Analogously the representation theory of dynamical quantum groups is related to special functions. We show the relation of a dynamical quantum group associated with an elliptic R-matrix and elliptic hypergeometric series in Chapter 3. In the last chapter we study the dynamical U{n) quantum group which is the dynamical analogue of the group of unitary n x n-matrices.

To a Lie group G we can associate two different Hopf algebras, i.e. a unital algebra with counit. comultiplication and antipode (or coinverse) satisfying certain relations. For the algebra of functions on the group, Fun(G), the group operations induce the structure of a Hopf algebra. Since the multiplication of Fun(G) is defined pointwise, it is a commutative algebra. We can also equip the universal enveloping algebra Z//(g) of the Lie algebra g corresponding to G with the structure of a Hopf algebra. The counit, comultiplication and antipode are completely determined by its values on g, and U{g) is a cocommutative Hopf algebra.

Roughly speaking quantum groups are deformations of algebras which have a Hopf algebra structure. One way to construct quantum groups is introduced by Drinfeld [12] and Jimbo [25], they studied examples of quantum groups which are deforma-tions of the universal enveloping algebra ^ ( g ) of a semisimple Lie algebra g. Faddeev, Reshetikhin and Takhtajan [18] studied quantum groups which are deformations of the algebra of functions on a group. They introduced a way to construct examples of quantiun groups from n^ x /i'^-matrices, this construction is nowadays known as the

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2 C H A P T E R 1. I N T R O D U C T I O N

FRST-construction (Faddeev, Reshetikhiu. Sklyanin. Takhtajan). The most interest-ing examples of such kind of quantum groups are constructed from matrices which are solutions of the quantum Yang-Baxter equation (1.2.1), since these quantum groups have suitable additional properties. For specific examples there are duality results between the two different types of quantum groups. In the setting of C*-algebras, compact matrix quantum groups were studied by Woronowicz [61]. More recently in the setting of C*-algebras, locally compact quantum groups were introduced by Kustermans and Vaes, see [39]. Quantum groups have applications in several fields of mathematics such as the theory of knots and braids, special functions and Knizlmik-Zamolodchikov equations, see e.g. [10], [13], [28].

Dynamical quantum groups are quantum groups with an extra parameter, the so-called dynamical parameter. These dynamical quantum groups are no longer Hopf algebras but so called ()-Hopf algebroids, which are a special case of the Hopf algebroids introduced by Lu [40], see also [57], [8]. f)-Hopf algebroids are no longer algebras over C but the scalars are extended non-trivially to M(,., the set of meromorphic functions on the dual space f)* of f). where () is a finite dimensional complex vector space. The theory of ()-Hopf algebroids was initiated by an example of a dynamical quantum group constructed from an elliptic solution of the quantum dynamical Yang-Baxter equation studied in [19], [20]. In [16], [17], Etingof and Varchenko developed the algebraic framework for these quantum groups. In this thesis we study dynamical quantum groujjs which can be seen as deformations of algebras of functions on a group. All examples we discuss in this thesis are constructed from solutions of the quantum dynamical Yang-Baxter (with or without spectral parameter) by the so called generalized construction, see §2.1.2. which is a generalization of the FRST-construction for quantum groups. These dynamical quantum groups have a natural self-duality related to the so called /?-matrix from which they are constructed, see Lemma 2.2.5. In [56], Stokman showed a relation between the quantum SL(2) group and the dynamical SL{2) quantum group via a dynamical twist. So far, this relation is only valid in the SL[2) case since a key ingredient in the construction of the dynamical SL{2) quantum group via a dynamical twist is the twisted coboundary element of Babelon, Bernard and Billey [2] in the form discovered by Rosengren [45] which is only explicitly known in the s[(2,C) case, see [9].

The simplest examples of (dynamical) quantum groups can be seen as deformations of the group SL{2, C), the group of 2 x 2-complex matrices with determinant equal to 1. 5 L ( 2 , C ) is a complex 3-dimensional Lie group. The matrix elements of its finite dimensional representations can be rewritten in terms of hvpergeometric series, see §1.1.1. When we restrict to the group SU{2). the group of unitary 2 x 2-matrices with determinant equal to 1, we obtain orthogonality relations for the matrix elements. These relations can be identified with discrete orthogonality relations for Krawtchouk polynomials. In Chapter 3, for a dynamical quantum group associated to an elliptic i?-matrix a related construction is carried out. For this elliptic quantum group, the matrix elements are expressible as certain elliptic hvpergeometric series and we obtain

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discrete bi-orthogonality relations for elliptic hypergeometric series.

In the following table the above discu.ssion is presented also including the interme-diate steps of the quantum SU{2) group and the dynamical SU{2) quantum group. The quantum SU{2) group will be discussed in Example 1.2.4, the dynamical SU{2) quantum group is presented in Example 2.1.24. These quantum groups are constructed from solutions of a quantum Yang-Baxter equation, see §1.2.1, which we indicate in the first column. The vertical arrows represent formal limit transitions, these limits will be discussed in more detail in Examples 1.2.4, 2.1.24 and Remark 3.1.6.

(functions on) SU(2) —» Krawtchouk oFi

T T

QYBE —> quantum SU(2) group -^ g-Krawtchouk 2(tii

T T

QDYBE -^ dynamical SU{2) quantum group -^ g-Racah 4(^3

T T

QDYBE spectral -^ elliptic U(2) quantum group —» ellip. hyperg. 12VI1

This chapter is organised as follows.

In §1.1 we recall the finite dimensional representations of the group 5 L ( 2 , C ) , and obtain orthogonahty relations for Krawtchouk polynomials when restricting to SU{2). In §1.1.2 we reformulate some of the results in the setting of Hopf algebras and we give the pairing between this algebra and the universal enveloping algebra of the Lie alge-bra s((2. C). In Chapter 2 we discuss a similar setting for dynamical quantum groups. In §1.1.3 we recall how a solution of the quantum dynamical Yang-Baxter equation (1.2.4) can be obtained from tensor products of the finite dimensional representations of 5 ^ ( 2 , C). This result is essentially due to Wigner [60]. Solutions of quantum Yang-Baxter equations are an imjjortant ingredient in the theory of (dynamical) quantum groups, in §1.2 we introduce the different types of quantum Yang-Baxter equations. From solutions of the quantum Yang-Baxter equation (1.2.1), called i?-matrices, we can construct quantum groups using the FRST-construction due to Faddeev, Reshetikhin and Takhtajan [18], see §1.2.2. A similar method introduced by Etingof and Varchenko [16], [17] can be used to construct dynamical quantum groups from solutions of the quantum dynamical Yang-Baxter equation (with or without spectral parameter), see §2.1.2. As an example we show how to construct the quantum SL{2) group, which can be seen as a deformation of the algebra of polynomials on the group 5 L ( 2 , C ) , from a solution of the quantum Yang-Baxter equation (1.2.1). In §1.3 we recall hypergeo-metric, basic (or q-) hypergeometric and elhptic hypergeometric series. This section is mainly meant to fix notations, for more information we refer to [23]. We conclude this chapter with an overview of the thesis in §1.4.

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1.1 T h e group 5 ^ ( 2 , C)

In this section we study the group SI,(2. C) and its representation theory, see also [58]. From the unitarity of certain finite dimensional representations restricted to SU{2) we obtain orthogonality relations for Krawtchouk polynomials. In §1.1.2 we reformulate some results for this example in terms of Hopf algebras and their (co-)representation theory, which is also the setting for studying quantum groups. We also show that the Racah coefficients or 6j-symbols of the group SL(2,C) satisfy the quantum dynamical Yang-Baxter equation (1.2.4).

1.1.1 Representations of SL{2,C) and Krawtchouk

polyno-mials

The group SL{2,C) is the group of all 2 x 2-matrices with complex entries with de-terminant equal to 1. Let V'^' be the (iV -I- l)-dimensional vector space over C of homogeneous polynomials of degree TV in two variables. We denote elements of V'^ by p ( a , 7 ) , this notation is inspired by the fact that V'^ is the analogue of the corep-resentatiou for the elliptic U{2) quantum group studied in Theorem 3.3.3. A basis of V^ is given by {e^jf^o := {a^~''l''}^=o- Then we can define an action of 5 L ( 2 , C ) on V'^ by

TT{g)p{a.^) = p{aa + C7, ba + d-y) G V^, ( l - l - l )

with g = ( " yi e SL{2.C) and p e V'^. The map TT : 5 L ( 2 . C ) ^ GLiV'^) is a ^c dj

homomorphism and is called a representation of SL(2, C) on V'^ .

We can explicitly compute the matrix elements f^,j of the representation V'' defined by 7r((?)e^ = '^n=o^kni9)^n ^ by calculating the action of an element g

-a 6^

, , G SL(2. C) on a vector of the basis,

7r(5)e^(a, 7) = {aa + nf-\ba -h ^7)''

E E 7 a^-'^-^b'-^c'd' a''-'-^f+'

1=0 j=o ^ ' ^-^

j=max{0,k+n— N)

^E"-'-"'v 2: (•::;) G)(^)'^"'->'

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can be rewritten as Krawtchouk polynomials, see Example 1.3.3 for its definition.

^ ' ^ a - ^ - ' ' - " 6 V 2 F i ( - n , -A- - . V ; - ^ )

n) ' '^ 'be'

n a-^-''-Wc''K\{k:-bc.N),

using a transformation formula for 2^^] 's in the second equality. For g G SU{2) we have ^nkid ') = I u I ^knid)- ^'^'*' follows hnmediately from the fact that a = d, b =

- c for a matrix g = (^^ J ^ e 5C/(2). Define { / f jf^^ = {(J^ e^Lo ^ a new basis for V'^ and define the matrix elements T^{g) by 'n'{g)fi^ = Yln=o'^kn{9)fn • We can equip V'^ with an inner product (•, •) such that {fk}k=o is an orthonormal basis. Then we immediately see that n{g^^)f^ = i^igYf^ for g £ SU{2), so the restriction of TT to SU{2) is a unitary representation, i.e. {TT{g){f^). fj^) = {f^.TT{g-'){fp) for all g e SU(2). For g € S't/(2), we have

/ f ( « , 7 ) = <9-'9)fk^{a,l) = ^(ff)M5) ƒ,'"(«, 7) = f^ ï f ï ^ T . ' K ^ ) / / ( a , 7 ) ,

which implies Yli=QTjl{g)T^^{g) = (5^^.. Using the explicit expressions of the matrix elements T^„{g) and denote 9 = [ i] ^'^ Set

Y, Kk{i- -be, N)K,{i; ^bc, N){ad)''-'{-bcy f^^) = f^] {adf{-bc)-'6kj,

using a = d,b = —c for g e SU{2). Since ad — bc = 1 these relations are equivalent to the orthogonality relations for Krawtchouk polynomials (1.3.4).

1.1.2 T h e Hopf algebra V(SL{2, C)) and duality

Instead of studying the group SL(2, C) itself we now look at the algebra of polynomials on 5'L(2,C), which we denote by 'P{SL{2,C)). This is a commutative algebra with pointwise multiplication. Translating the group properties of SL{2. C) to 'P(SL{2, C)) makes V{SL(2,C)) into a Hopf algebra. In Example 1.2.4 we study the quantum SL{2) group which is a non-commutative deformation of this algebra. In the limit g ^ 1 the results for the quantum SL{2) group give back the V(SL{2. C))-setting.

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6

C H A P T E R 1. I N T R O D U C T I O N

The algebra V{SL{2,C)) is generated by four coordinate functions a, jJ, 7 and ö which are defined by

a{g) = a. 0{g) = b, 7(5) = c. 6{g) = d, (1.1.2)

for <; = . J. Since det(g) = 1 for all g G SL{2.C). the generators satisfy the relation aS — /i7 = 1. For this algebra we have the identification V{SL{2,C)) (8) r{SL{2, C)) ~ P{SL{2. C) X SL{2, C)).

For a finite group G the multiplication of the group induces a homomorphism A : C{G) —> C{G) ® C{G) on the algebra of continuous functions by A{f)(g, h) = f{gh) for all ƒ G C{G) and all g. h € G. This map automatically satisfies the coassociativity property (A ® Id) o A = (Id (81 A) o A. Similarly the unit element eg of the group G induces a homomorphism e : C{G) —» C defined by e{f) = f{ec,) which satisfies (e (gi Id) o A = Id = (Id 0 e) o A. The group inverse induces a map S : C{G) —> C{G) defined by S{f){g) = f{g~^) which satisfies m o ( 5 0 Id) o A = rjoe = mo [Id® S) o A where m : C{G)®C(G) -^ C{G) is the multiphcation in the algebra C{G) and the unit map r/ : C —> C[G) is defined by ri(z) = ZIQ. with If; the map identically 1 on G. In general the antipode is an algebra anti-homomorphism, since C[G) is a commutative algebra S is also a multiplicative map in this special case. The maps A, e and S are called the comultiplication, counit and antipode (or co-inverse) of C[G) respectively. The algebra C[G) with these extra structures is an example of a Hopf algebra. For an introduction to Hopf algebras and quantum groups we refer to [28], [10]. In Chapter 2 we give the definitions for h-Hopf algebroids which in the trivial case that \) — {0} reduce to Hopf algebras.

In the example of •p(5L(2,C)), the comultiplication A, counit s and antipode 5 can be explicitly given by

A ( a ) = Q(8)a-h/?(8)7, A{l3) = a® (5-^ (3 ® 5

A(7) = 7 ( 8 ) Q - h J ® 7 , A((5) = 7 (g)/3-h (5 ® J, fi 1 -^^

K ° ^ ( ; ; ) < : ^ ( w f )

-The representations (1-1.1) of the group 51/(2, C) on V^ induce the corepresentations p-.V'^ ^ V(SL{2. C)) ® V^ of ^ ( 5 1 . ( 2 . C)) on V"^' by

In general, a (left) corepresentation of a Hopf algebra A on a vector space V is a map p:V ^ A®V such that (A 0 Id) o p = (Id ® p) o p and (s Cg) Id) o p = Id.

We define a ^-operator, i.e. an antilinear antimultiplicative involution, on V(SL{2. C)) by

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on the generators. This *-operator satisfies (*®*)oA(a) = Ao*(a) and (so*)(a) = e{a) for all a £ V{SL{2.C)). We can also describe this ^-operator on V{SL{2,C)) by f{g) = f{e{g)) where Ö is the involution on SL{2.C) defined by 0(g) = (ff*)"'. The algebra 7-*(5L(2,C) with this ^-operator can be seen as the real form since SU{2) = { . g G 5 L ( 2 , C ) : p * ( 5 ) = R ^ f o r a l l p } .

Since the representation TT restricted to SU{2) is unitary, this implies that for g e SU{2) we have T^^ig)* = S{T^„{g)). With this property we call p a unitary corepresentation, and the unitarity of p gives orthogonality relations for Krawtcliouk polynomials. In §3.3 we define the analogue of this corepresentation for the ellip-tic U{2) quantum group, from which we obtain discrete biorthogonality relations for certain elliptic hypergeometric i2Vii-series.

There exists a natural notion of duality between V{SL(2,C)) and the universal enveloping algebra W(sl(2,C)) of the Lie algebra s[(2,C). The Lie algebra sl(2,C) is the complex vector space of the 2 x 2-complex matrices with trace equal to 0, where the Lie bracket is given by [X. Y] = XY — YX using matrix nndtiplication on the right hand side. In general, the universal enveloping algebra Z//(0) of a Lie algebra g is the quotient of the tensor algebra ©„>o 0®" and the two sided ideal generated by the elements x ^ y — y ^ x — [x,y] for all x,y E Q. So W(s[(2,C)) is generated as an algebra by all elements X € s[(2, C) with the relation X (^Y - Y (^ X = [X,Y] for all X. Y G Q. W(s[(2,C)) is a Hopf algebra with counit e, comultiplication A and antipode S defined by e{X) = 0. A{X) = 1 0 X + X ® 1 and S{X) = -X for all X € s[(2, C) and e, A extended as algebra homomorphisms and S as an algebra anti-homomorphism. The universal enveloping algebra U{sl{2, C)) is a cocommutative Hopf algebra, i.e. P o A = A where P is the flip operator define by P{Xi^Y) = Y^X for all X, r G W ( S [ ( 2 , C ) ) .

There exists a pairing (•, •) : U{5l{2.C)) x P(5L(2,C)) -^ C defined by

f{exp{tX)). (1.1.6)

(=0

for all ƒ G V{SL{2.C)), X € s[(2, C), where exp is the exponential function exp : s[(2.C) -^ SL{2X) which is given by exp(A') = E r = o ^ V ^ ' ! - and the pairing is extended to Z^(s((2,C)) x P{SL{2.C)) by (1.1.7a). This pairing behaves nicely with respect to the structures of the Hopf algebras, see Definition 1.1.1.

In general the duality between Hopf algebras is defined in the following way.

D e f i n i t i o n 1.1.1. A pairing between two Hopf algebras A and U is a bilinear form {•,•) -.U X A^C such that

{XY,a) = {X®Y,A{a)), (X,a6) = ( A ( X ) , a ® 6), (1.1.7a)

(1, a) = e[a), {X, 1) = e[X). (1.1.7b)

<^-f^=7t

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and

( 5 ( X ) , a ) = ( X , 5 ( a ) ) , (1.1.8) for all X,Y G hi and all a. h £ A. If m addition a ^-operator, i.e. an antilinear

antimultiplicative involution such that {*(gi^)oA = Ao* and * o s = e o * , is defined on A andU, we also require

{X\a) = {X,S{a)*), (1.1.9)

for all X eU and all a e A.

If the pairing is non-degenerate, i.e. {X, a) = 0 for all X E U (respectively for all a G A) implies a = 0 (respectively X = Q). A and U are said to be in duality.

R e m a r k 1.1.2. (i) In (1.1.7a) we use that the pairing can be extended to a pairing on A^ A and U ®IA hy {a®h,u®v) = (a,u)(6, v) for all a, 6 G «4 and all u, v £U. (ii) For every two Hopf algebras U and A there exists the trivial pairing given by {X, a) = £{X)e{a) for a\\ X £ U. a G A. We are interested in cases where there exists a non-trivial pairing.

By defining a *-operator on sl(2, C) by taking the adjoint and extended to W(sl(2. C)) antilinearly and antimultiphcatively we obtain that (1.1.9) is satisfied for the pairing (1.1.6). We can define a left and right action of sl(2.C) on V{SL(2X)) by

\x-fM=i

figexpitX)) [f-X]{g)

d

Jt

f(exp{tX)g), (1.1.10)

respectively. This gives for every X G s[(2,C) an action by a first order differential operator. In general, if A and U are paired as Hopf algebras we can define a left and right action of W on ^ by

X • a = (Id ® (X, •))A(a), a-X = ({X. •) ® Id)A(a), for all X eU and all a G >l respectively.

In Chapter 2 we extend this notion of duality to the setting of fj-Hopf algebroids, see Definition 2.2.1 and we develop the theory of actions for f)-Hopf *-algcbroids. Note that there does not exists a trivial pairing for f}-Hopf algebroids analogous to the trivial pairing for Hopf algebras. So far we cannot lift the duality property for 'P{SL{2.C)) and U{sl(2.C)) to the elliptic level. That is, we cannot prove that the elliptic U{2) quantum group is paired with some deformation of the corresponding universal enveloping algebra. In Theorem 3.2.2 we prove that the elliptic U(2) quantum group and its coopposite are (non-trivially) paired as f)-Hopf *-algebroids, which do give a kind of self-duality.

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1.1.3 Racah coefficients and the dynamical Yang-Bcixter

equa-tion

In this subsection we show how to obtain a solution of the quantum dynamical Yang-Baxter ecjuation (1.2.4) using the representation theory of SL(2, C), this result is essentially due to Wigner [6Ü].

Recall the representation yr of 51/(2, C) defined on the space V'^ in §1.1.1. We can define an action of 5L(2, C) on the tensor product V'^^ <g> V^^ and the direct sum yNi 0 YN2 |3y -iT{g){vi ® V2) = T^{g)vi ® T^{g)v2 and 'ïï{g){vi ® «2) = Tr{g)v^ ® T^{g)v2 respectively, for g G 5 L ( 2 , C ) , v\ e V^^ and V2 € V^'^. An intertwmer between two representations (IT, V) and {p, W) of a group G is a linear map T : V ^ W such that T o n{g) = p{g) o r for all g & G.

The representations V^ are in some sense all representations. Namely, every finite dimensional irreducible representation of 5 L ( 2 , C ) , i.e. with no non-trivial invariant subspaces, is isomorphic to V'^ for some N £ N.

T h e o r e m 1.1.3. V'^, N £ Z>o, is a complete set of finite dimensional irreducible representations of SL{2,C) up to isomorphism. Any finite dimensional representation of SL(2,C) is isomorphic to a direct sum ofV^ 's.

From Theorem 1.1.3 we obtain that the tensor product V^'-^V'^^ is isomorphic to some direct sum of the V^'s. Explicitly, the Clebsch-Gordan decomposition for the repre-sentations V, A^ G N. of 5 L ( 2 , C ) , is given by V^' ® V^^ = ©™'^'^''^'' yN,+N2-2s^ Then the Clebsch-Gordan coefficients (or 3j-symbols) Cj^i'^^^" are defined by

s.n k+l=s+n

these coefficients are the matrix coefficients of the intertwiner. this expression turns out to be a single sum by considering the action of a diagonal matrix.

In this case the Clebsch-Gordan coefficients can be written as certain 3F2 hypergeo-metric functions, see e.g. [58]. Define the intertwiner J, ; V^' ig) V^^ -^ YNI+N2-2S by

Js{ek' ^ ef^) = Cj^if'-'e^^+''^-'^ iU'+ I = s + n and zero otherwise.

The Racah coefficients (or 6j-symbols) are usually defined as tlie coefficients of the intertwiner between two different basis of the three-fold tensor product of representa-tions, namely of V'"' ® {V^^ ® V^^) and (V^' 0 V^^) ® V^\ See also [49], where Rosengren studies 6j-symbols by an elementary approach not using the underlying (quantum) groups explicitly. Since the flip operator P : V^'- (8> V^^ —> V'^^ igi V^^ defined by P{v (g) w) = w ^ v, ïor v e V^^, w € V'^^ is an intertwiner we can also define the Racah coefficients ƒ?"" by

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with Vi £ V'^', V2 e V'^'-', V3 € V'^^. So these Racah coefficients are defined by the relation V^^ ® {V'^^ ® V^^) ~ V^^ ® (V^^' ® V''^^). Then using V^' ® (V^^ 8) {V'^^ (g V/A'.i)) ^ v''^» (g) (\/'^2 ^ ('[/Ari ^ v'^")), which can be seen as the symmetry for Wigner's 9ji-coefficients, we obtain a relation for the Racah coefficients which turns out to be equivalent to the quantum dynamical Yang-Baxter equation (1.2.4). We can compute, using (1.1.11) three times,

Jp(l'l® J,(l'2 $5 Jr{V3 ® V4)))

= Y^ Rll{NuN2.Ni + N^-2r)Jy{v2®Uv,®Jr{v3®V4)))

p+q=x+y p+q—x+y a+b—x+r X Jy{v2 ® Jh{v3 ® Ja{Vi ® V4)))

= L E E «pT(^i'^2,A^3 + iV4-2r)iïf,(yVi,iV3,7V4)

p+q=x+ya+b—x+rc+d=y+b

X Rli{N2, N3. iVi + 7V4 - 2a) Jd(t;3 ® Je(w2 ® ^a(vi ® Vi))), for Ui G y'*''. Similarly we get, by applying (1.1.11) to the last part first,

Jp{Vilg)Jq{v2(S)Jr{V3<E)V4)))= E E E ^9r(^^2,A^3,A'4) k+l=q+r m+n=p+l i+j—m+k

X 7?7(yVi, 7V3, ^ 2 + ^ 4 - 2k)R'^,{Ni.N2.7V4) J„(i.3 ® Jj(w2 ® Mvi ® 1^4))), for Uj G 1/^'. These two expressions being equal gives

E E E R?qiNuN2,N3 + N,-2r)

p+q^x+ya+b=x+rc+d=y+b

X Rf,{NuN3, Ni)Rli{N2. N3. m + N^ - 2a)

= E E E Kri^^, N3,N,)

k+l~q+r m+d—p+l a+c—m+k

X R^i''{NuN3, N2 + N4- 2k)R"^,{N,.N2, N4),

where the summations are essentially triple sums. This relation can be reformulated in terms of the quantum dynamical Yang Baxter equation (1.2.4). where R{\) : V ^ ' ® V'^2 _^ y M ^ yN2 ig defined by

i?(A)ef ® ef^ = ^ R'^l'iN,, iV2, A - A^i - TVj + 2A- + 2 t ) e f ® e^^

and the weight of e^ is A'^ — 2s. So the Racah coefficients of the representations V'^ of SL(2, C) give a solution to the quantum dynamical Yang-Baxter equation. In this way we constructed a solution of the quantum dynamical Yang-Baxter equation from the representation theory of the group 5 L ( 2 , C ) . In §3.4.2 we construct a solution of the quantiun dynamical Yang-Baxter equation with spectral parameter using the representation theory of the elliptic U{2) quantum group.

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1,2 Quantum groups

Most interesting (dynamical) ciuantum groups, such as deformations of an algebra of functions on a group, are constructed from solutions of Yang-Baxter equations. In §1.2.1 we introduce the different kind of quantum Yang-Baxter equations and give an idea what kind of quantum groups are related to solutions of these equations. We give the FRST-construction, which assigns to any solution of the quantum Yang-Baxter equation (1.2.1) a bialgebra. In Example 1.2.4 we define the quantum SL{2) group which is constructed from the /^-matrix (1.2.2) by applying this FRST-construction. We show that this quantum grouj:) can be seen as a deformation of the algebra of functions on the group SL(2, C) and indicate how to obtain orthogonality relations for g-Krawtchouk polynomials.

1.2.1 Q u a n t u m Yang-Baxter equations

In this subsection we introduce the different kinds of quantum Yang-Baxter equations, and give some solutions. We shortly discuss the relation of solutions of these quantum Yang-Baxter equations to (dynamical) quantum groups. In Example 1.2.4 we recall some results for one of the easiest examples of a non-commutative, non-cocommutative quantum group: the quantum 51/(2) group, which is a deformation of the algebra V{SL{2,C)) of polynomials on 5 L ( 2 , C ) . In Chapters 2. 3 and 4 we study examples of dynamical quantum groups in detail.

The Yang-Baxter equation appears in many different subjects of physics, such as statistical mechanics and integrable models, see e.g. [4], [38], [24]. Solvability of so called vertex models is related to the quantum Yang-Baxter equation with spectral parameter, where as solvability of face models is related to the quantum Yang-Baxter equation with dynamical parameter.

Quantum Yang-Baxter equation

Let F be a complex vector space. For a linear map R : V0V^V®Vwe introduce the notation R^'^{u'S) v ^ w) = R{u^v)iS>w for all u, v, w 6 V. Analogously we define

ƒ?'•', 7?^^. A linear endomorphism R: V®V^V®V t h a t satisfies the quantum Yang-Baxter equation

^ 1 2 ^ 1 3 ^ 2 3 ^ ^23^13^12^ ( ^ 2 . 1 ) in End(V ^V ^V) is called an i?-matrix.

R e m a r k 1.2.1. The quantum Yang-Baxter equation (1.2.1) is related to the braid

group relation. The braid group on A' strands is the group B^ with A — 1 generators 6 i , . . . , ö/v-i with the relations

b,bj = hjb,. for \i - j \ > 1, ö,fe,+i6, = ö,+i6,öi+i.

If 6, is the change of the strands i and i + 1, then the second relation which is also called the braid relation represents the fact that the order of three strands can be reversed by changing two neighboring strands in each step in two ways.

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Let P: V($V^V®V the flip operator defined by P{u<S) v) = i' (g) u for all u, (- e V. For a linear map R:V^V ^V<S)V define the operator R' : V^^ -^ V^'^ by R' := P"+^ o R'-'+\ using the notation as introduced above. U R : V (^V -^ V (g)V satisfies the quantum Yang-Baxter equation (1.2.1) then R^ satisfies the braid relation

R'R'+'R' = R'+^R'R'+\

as identity in V'^'^. Since the relation R'W = RJR' for all \i - j \ > 1 is trivially valid. we can see R', i = 1 A^ — 1, as a representation of the generators of the braid group ö^'. See e.g. [28], [10] for more information.

It is easy to check that R = Id and R = P, with P the flip operator, are sohitions of the quantum Yang-Baxter equation (1.2.1).

Let V = Cei ® Ce_i be a two-dimensional vector space. Let 0 < ^ < 1 be a fixed constant. In the basis ei ® ei, ei 0 e_i, e_i (gi ei, e^i 0 e_i the matrix R^ given by

/(? 0 0 0 \

R,- () 0 1 0 (1-2.2) \ 0 0 0 qj

is a solution of the quantum Yang-Baxter equation. In Example 1.2.4. we construct the quantum 5L(2) group by applying the FRST-construction to this solution. This matrix in addition satisfies the Hecke condition, that is R'^ = (q ~ (/^')7?q + 1.

Quantum Yang-Baxter equation with spectral parameter

Let V be a complex vector space. Introduce the notation R^'^{z){u^v<^w) = R{z){u<^ v) ^ w for a linear map /? : C —> E n d ( y ® V). A meromorphic function i? : C —» E n d ( V ® V) that satisfies the (juantum Yang-Baxter equation with spectral parameter

R''{z,/z,)R''{z,/z,)R''{z2/z,) = R''(z2/zs)R''(z:/z,)R''\zJz,). (1.2.3) for all Zi, Z2, z^ € C is called an i?-matrix. Rational solutions of the quantum Yang-Baxter equation are related to so-called Yangians, which are a type of infinite-dimensional quantum groups, see e.g. [10, Chapter 12].

Quantum dynamical Yang Baxter equation

Let f) be a finite dimensional complex vector space viewed as a commutative Lie algebra, with dual space \)*. Let V = 0 Q g ^ . K, be a diagonalizable [)-niodule, i.e. V„ = {v^V •.H-v = a{H)v. for ah H 6 i}}. For a function i? : ()* ^ End{Vi^V) we introduce the notation i?'^(A - /;'^')(w (g) f (g) w) = R{X - ii){u®v) (gw for ah A G f)*, and all u. r e V. w G 1],. A function ƒ?:[}* ^ End(l^ (g) V) is called [}-invariant if it commutes with the [)-action on V ($V i.e. R(X){Va <g Vp) C 0 ^ ^ ^ ^ ^ ^ ^ Vy (g Vs for all A G [)*. A meromorphic function i? : f)* —» E n d ( y (g) V) which is (^-invariant and satisfies the quantiun dynamical Yang-Baxter equation

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for all A € i)*. is called a (dynamical) i?-matrix. The parameter A is called the dynamical parameter.

In Example 2.1.24 we study the dynamical U{2) quantum group related to the solution (2.1.13) of (1.2.4), which is a solution in the case that () ~ ()* i ; C and V is a 2-dimcnsional vector space. The matrix (2.1.13) can be seen as a generalization of (1.2.2). To be precise, taking the limit A -^ —oo in (2.1.13) we get back (1.2.2). In Chapter 4 we study the dynamical U{n) quantum group constructed from the solution (4.1.1) for f) ^ ()* ~ C" and V is an ?i-dimensional vector space. For both solutions the elements of the i?-matrix are trigonometric functions of the dynamical parameter.

Quantum dynamical Yang-Baxter equation with spectral parameter

Let f) be a finite dimensional complex vector space viewed as a commutative Lie algebra, denote by f)* its dual. Let V = ®„g[,. Va be a diagonalizable [)-module. For i? : (l* X C —> End(V' ® V) we introduce the notation i?'^(A - ^<^>, z){u ® v ^ w) = E{X - ii.z){u ® i>) ® w for all A G ()*, all z e C and aU ?;.ii e V. w e V^,. A meromorphic function R: f}* x C —^ E n d ( V ® F ) which is ()-invariant and satisfies the ciuantum dynamical Yang-Baxter equation with spectral parameter

i?''^(A - h^'KzJz,)R''\X,z,/z3)R'\X - h^'\z,/zs)

for all A € f)' and all zi, Z2, 23 € C is called a (dynamical) i?-matrix. In Chapter 3 we study the elliptic U{2) quantum group constructed from the solution (3.1.1) of this Yang-Baxter equation for f) ~ f|* :^ C and V is a 2-diniensional vector space. The elements of this /?-matrix are elliptic functions involving the dynamical and spectral parameter. This /?-matrix can be seen as a generalization (up to gauge ecjuivalence, see §2.1.2) of the dynamical matrix (2.1.13). see Remark 3.1.6 for more details.

R e m a r k 1.2.2. A function ƒ : C" —> C is called holomorphic if for all XQ € C" there

exists a neighborhood UJ•^^ and a power series X^^Q'*"(-'" " -^o)" ^^^^^ converges to ƒ on Uxo- A meromorphic function on C" is a function which can be written locally as the quotient of two holomorphic fimctions.

For later reference we introduce the matrix elements of an /?-matrix and refornuilate the quantum dynamical Yang-Baxter equation with spectral parameter in a relation for these matrix elements. Of course we can similarly obtain a reformulation in terms of matrix elements for the other types of quantum Yang-Baxter equations.

Let X be a finite index set and w : X -^ \)*. Let {ex}xex be a homogeneous basis of V, such that Cx G V^(x). Write Rly(X, z) for the matrix elements of R,

R{X,z){ea(8)ek)= ^ i?f^(A, 2)e^ ® e^. (1,2.6) x.yeX

Applying the left and right hand side of (1.2.5) to ÊQ (g) e;, 0 e^ and identifying the coefficients of e^ ® e/ ® Cg gives the quantum dynamical Yang-Baxter equation in

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terms of the matrix elements of R

J2 Kfi^ - ^(9)^ ZI/Z2)R:'^{X, Z,/Z3)R%{X - Aa). z,/zs) wxy

= ^ R];{X, Z2/z,)RZ(X - Ay)^ z,lz,)Rf^{X, z,lz^). wxy

The formal limits between the different types of Yang-Baxter equations also imply limits between the constructed quantum groups. From the elliptic GL{2) group of Chapter 3 we can obtain the dynamical GL{2) quantum group of Example 2.1.24, see Remark 3.1.6 for more details. From this dynamical GL{2) quantum group we can obtain the quantum GL{2) group which is a deformation of the algebra of polynomials on the group GL{2). see Example 2.1.24. In this way also the elliptic GL[2) group can be seen as a deformation of the algebra of polynomials on the group GL{2). Roughly speaking, these limit transitions also survive in the representation theory of these quantum groups.

1.2.2 FRST-construction

Many interesting examples of non-commutative, non-cocommutative Hopf algebras are constructed from solutions of the quantum Yang-Baxter equation (1.2.1). From an /?-niatrix we can construct a bialgebra, i.e. a unital algebra A with comultiplication /\ : A ^ A® A and counit £ : ^ -^ C satisfying (A (g) Id) o A = (Id ig) A) o A and (e i8> Id) o A = Id = (Id ® s) o A, using the FRST-construction. This construction was introduced in [18]. As an example we discuss some aspects of the quantum SL{2) group, which is a deformation of the algebra 'P{SL{2. C)) of polynomials on SL{2, C).

Let 7? £ A/„2(C) be an arbitrary non-singular matrix. We equip C" with the standard basis {ej}"^j, i.e e^ is the z-th unit vector of C , and C" ® C" with the basis e\ 0 Ci, ei 0 6 2 , . . . , e2 (8) ei e^ ® e„ using the lexicographical ordering. Let AR be the unital complex associative algebra generated by the elements {iy}"j=i subject to the relations R{T ® I){I ig T) = (ƒ » T){T ® I)R where T = {tij)ljj^ and I the n X rj-identity matrix. These relations can be rewritten as

u n

J2 Kct^btyd = Yl K%yta.~ for all a, &, c, d e {1 /),}, (1.2.8) x,y=\ x.y~l

where the matrix elements iïj^ of R are defined by the analogue of (1.2.6). The counit e : AR —» C and comultiplication A : AR -^ AR (g) AR are defined on the generators by

n

e{Uj) = (5y, A(ty) = ^ i i / t ® tkj, (1.2.9)

and extended as algebra homomorphisms. Then we can easily check that AR is a bialgebra. It suffices to check that (1.2.8) remains vahd using (1.2.9) and £, A being extended as algebra homomorphisms.

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This construction works for general matrices R but R being a solution of the quantum Yang-Baxter equation (1.2.1) is the most interesting case. In §2.1.2 we discuss a generalization of the FRST-construction which assigns to a dynamical i?-matrix a f)-bialgebroid.

R e m a r k 1.2.3. The algebra of polynomials on A/(2,C), the space of 2 x 2 complex

matrices, can be constructed by the FRST-construction from the i?-matrix R = I where I is the 4 x 4-identity matrix. Then the relations (1.2.8) represent the fact that the generators tu = a, tu = 0, ^21 = 7 and ^22 = <5 commute. The algebra P ( 5 L ( 2 , C ) ) of polynomials on 51/(2, C) is the quotient space of this space and the two-sided ideal generated by det — 1 = aS — 0"/ — 1.

E x a m p l e 1.2.4. In this example we study the quantum SL{2) group, constructed from the solution (1.2.2) of the quantum Yang-Baxter equation (1.2.1). The quantum 5L(2) group can be seen as a non-commutative deformation of the group of polyno-mials on 5 L ( 2 . C ) introduced in the previous section. In the limit 5 —> 1. we get back the algebra V(SL{2.C)) of polyncjmials on 5 L ( 2 , C ) from the quantum SL{2) group. Let 0 < (7 < 1 be fixed. Let ^ , ( i \ / ( 2 ) ) be the algebra with generators t\i = a, t\2 = P, ^21 = 7 and /:22 = ^ and defining relations given by (1.2.8) using the i?-matrix (1.2.2). The 16 defining relations (1.2.8) can be reduced to the 6 independent relations 0a = qaP, 7 0 = 507, 60 = qp6, S-, = q^iS ( 1 2 10) /?7 = 7/?, no — Sa = (g~' — q)P^.

This algebra can be seen as the quantum analogue of the algebra of polynomials on M{2). The quantum determinant det = aS — q^''Pj is a central element of the algebra Tg{AI{2)), i.e. det commutes with all elements of J^g{M{2)). Let / be the two-sided ideal generated by det — 1. Define the quantum 5L(2) group as the quotient algebra Tq{M{2))/1. and denote it by !Fq{SL{2)). Then the counit e, the comultiplication A and the antijjode 5 defined on the generators by

A ( Q ) = a ® a - h / 3 ( » 7 , A[d)=a®0 + d®5 A(7) = 7(8)0-1-(5(817, A((5) = 7 ( 8 0 + 5(8 5,

equip J^q{SL{2)) with a Hopf algebra structure. Note that for the quantum determi-nant we have A(det) = det (8 det and £(det) = 1. It can be easily seen that in the limit g —» 1 we indeed get back the commutative algebra 7-'(5L(2,C)). The quantum SL(2) group can be seen as part of the dual space of the quantized universal envelop-ing algebra of sl(2,C), which is a deformation of the universal envelopenvelop-ing algebra of 5l(2,C), seee.g. [28].

We can define a corepresentation ol Tq{SL[2)) on a ciuantum analogue of the space of homogeneous polynomials in two variables V^ of §1.1. Let V^ be the space of homo-geneous polvnomials of degreee A' in two non-commuting variables a. 7 which satisfy

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the relation ja — qa^. Then a basis for V^ is given by {e^j^^o ~ l*^^ *^7*}fc=o- For iV = l,

7r(Q) = a ® a +/3(8i7, 7r(7) = 7 0 a + (5 ® 7, (1.2.11) defines a corepresentation of Tq{SL[2)) on V^. Note that in the notation Q, 7 are

generators of !Fq(SL{2)) or variables of polynomials in Vl^. In fact, n is just the comultiplication restricted to V^^ where we see V^^ as a subspace of !Fq{SL[2)). Since the comultiplication restricted to V^ satisfies /S.{V^) C Tg{SL{2)) » V^^, it defines a corepresentation of J^g{SL(2)) on Vl^. The matrix elements of this corepresentation can be written as g-Krawtchouk polynomials in non-commuting variables using the (j-analogue of the binomial theorem. Then from the unitarity of the corepresentation we obtain orthogonality relations for these polynomials, see e.g. [36]. For more results on the relation with q-hypergeometric series and this quantum SL{2) group see e.g. [30], [31] and references therein.

1.3 Special functions

As for (solutions of) the cjuantum Yang-Baxter equation we also have a division for special functions into hypergeometric series, basic (or g-)hypergeometric series and elliptic hypergeometric series. This classification is analogous to the classification of meromorphic functions in the Weierstrass' theorem, see e.g. [44] for a proof of this theorem.

T h e o r e m 1.3.1. Let P{a,b,c) be a non-zero polynomial in three variables. A mero-morphic function ƒ on the complex plane which satisfies the relation P{f(u), f{v), f{u-\-v)) = 0 is either a rational function of z, a rational function of e" for some constant c or an elliptic function (i.e. doubly periodic) of z.

Definition 1.3.2. A series J2T=o'^i^ ^^ called

(i) hypergeometric if Ck+i/ck is rational in k, for all k.

(ii) basic (or (7-)liypergeometric if Ck+i/ck is rational in g*'. for all k.

(Hi) elliptic hypergeometric ifc^+i/ck is an elliptic function ofk. where k is considered as a complex variable.

In this section we introduce the general notation for hypergeometric, basic hy-pergeometric and elliptic hyhy-pergeometric series, we use the notation of Gasper and Rahman [23]. See also [29] for the Askey-scheme, an overview of hypergeometric and basic hypergeometric orthogonal polynomials and its limit transitions.

1.3.1 (Basic) hypergeometric series

In this subsection we give the general notation for hypergeoiïietric and basic hyperge-ometric series.

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We introduce the shifted factorial

(a)o = l, (a)n = a(a + l ) - - - ( a + n - 1 ) , n e { l , 2 , . . . } . (1.3.1) Also denote (oi ak)^ = (ai)„ • • • (flfc)„. For a hypergeometric series J2^^o'^n ^'^^ quotient of two subsequent terms is rational in the sunnnation index, so in general we can write

Cn+^ ^ ( n + Q i ) • • • ( » . + « , ) ;

Cn {n + bi)---{n + b,){n + l)' The hypergeometric series rFs is defined by

rFs{au....ar\hi &,:2) = Y^ , . ' , \ " - 7 ' (1.3.3)

for bi bg ^ Z<o. If for some i, «j = —k with A' G N then r.f s is a finite sum and thus it is a polynomial in .:. Otherwise, the radius of convergence of the hypergeometric series is given by oo if r < s, 1 if r = s + 1 and 0 if r > s + 1. Every hypergeometric series can be written as a multiple of some

rF^-E x a m p l e 1.3.3. In §1.1 we showed that the matrix elements of a special represen-tation of the group S'L(2,C), the group of 2 x 2 complex matrices with determi-nant 1, can be written as Krawtchouk polynomials. These polynomials are given by Kn{x\p,N) = 2Fi{-n.-x;-N\\lp) for n € { 0 , 1 , . . . , iV}. The set {A'„(x;p, n)}^^o of polynomials of degree n in x is orthogonal with respect to a finite discrete measure. The orthogonality relations are given by

Y, A-,„(x;p, N)K^(x:p^ N) {^^^ f (I - pf-' = d„„(l - p ) > - " {^^^ , (1.3.4)

for 0 < p < 1. See also [29].

In the general notation of basic hypergeometric series we use the so-called (/-shifted factorial. We always assume 0 < g < 1. Define the q-shifted factorial by

n - l

(a;g)o = l, (a;g)„ = J ] ( l - a g ' = ) , n G { 1 . 2 . . . . } . (1.3.5)

A-=0

Again we denote (aj «*;(?)„ = (ai:g)„ • • • (OA-: g)„. Since l i m , | i ( l - g ' ' ) / ( l - < / ) = a we have \\Ti\g^i{q'^;q)nl{\ — g)" = (a)„. For basic hypergeometric series Yl^=a'^n the quotient of two subsequent terms is rational in g", so in general we can write

c„+i _ ( l - » i g " ) . - - ( l - a w " ) ( - g " ) ' + ^ - ' - z , . c„ ( l - 6 , g " ) - - - ( l - M " ) ( l - g " + ' ) • ^ • • ' Then the basic hypergeometric series r4>s is defined by

, . , . ^ . ^ ( « 1 a . : g ) n ( ( - l ) " 9 H " - " ) ' + - - - c " r(ps(au---,ar;bi,...,bs;q,:) = ^ - r r — ^ ^ . (1.3.7)

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18

C H A P T E R 1. I N T R O D U C T I O N If for some i, ai = q~^, with A' G N then the series terminates. Otherwise, the radius of convergence of the basic hypergeomctric series is given by c» if r < s, 1 if r = s + 1 and 0 if r > ,s + 1. In the limit g t 1 we have the formal termwise limit transition

l i m , 0 , ( g ' " , . . . , ( / ' " ; ^ ' " q''^:q,{q - 1)'+'-'-z) = rFs{ai a,; 6, b,;z) (1.3.8)

E x a m p l e 1.3.4. In Example 1.2.4 we indicate that the matrix elements of certain

corepresentations of the qnantiuii SL{2) group can be written as g-Krawtchouk polyno-mials. These polynomials defined by K„(q^^;p, N\ q) = 3(j)2{q~^, q~^. —pq"; q'^, 0; q, q), n € { 0 , 1 , . . . , A^} are orthogonal with respect to a finite discrete measure. Their or-thogonality relations are given by

TK„,{q-^;p,N-q)K„{q-':p,N-,q)^^-^^{-pr-(-p,q '^:q)n 1+pq^"

for p > 0. In the limit q 1 I we get back the Krawtchouk polynomials of Example 1.3.3:

lim Kn(q":p, N; q) = K„{x; l / ( p + 1), N).

Note that there are more (/-analogues of the Krawtchouk polynomials, see also [29].

1.3.2 Elliptic hypergeometric series

Elliptic analogues of very-well-poised hypergeometric series were introduced by Frenkel and Turaev [22] as elliptic 6j-symbols. Spiridonov and Zhedanov [55] re-obtained these bi-orthogonality relations in a different context, for more results on elliptic hypergeo-metric series see also [27], [46], [47], [51]-[54], [59]. We stick to the notation of Gasper and Rahman [23, Chapter 11].

The modified Jac.obi theta junctions arc defined by

r

9{Z) = {Z,P/Z-PU.

zeC\{0}. dim ar) = l[e{a,), (1.3.9)

where we assume 0 < p < 1. For later use we note that theta functions satisfy 0(pz) = 0(z^^) = —z~^6{z) and the following addition formula

6{xy.x/y, zw.z/w) = Bixw.xjw, zy,z/y) + {z/y)6{xz,x/z,yw,y/w). (1.3.1Ü) The elliptic shifted factorials are defined by

n-l k {a;q,p)n = J\0{aq'), ( o i , . . . . a t ; g . p ) „ = ]^(a,:<?.p)„, (1.3.11)

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for n = {(). 1 . 2 . . . } . The parameter q is called the base and p is called the uome of the factorial. Note that since limp^oö(ö) = (1 — a) we have limp_o(a; g,p)„ = {a:q)„. D e f i n i t i o n 1.3.5. Tlie theta hypergeometric series with base q and name p is (for-mally) defined by

r+\Er{a] ar+\-hi...,br\q.p,z) = y -— — 2 " . (1.3.12) ^ (q.bi,....br;q,p)n

In this thesis we only deal with terminating series, so we have no convergence problems. In general this series is only defined formally and you have to be very careful about the convergence properties, see [23] for more details. It turns out t h a t this theta hypergeometric series is an elliptic hypergeometric series under an extra condition, the elliptic balancing condition given by

aia2---ar+\ = qbib-z • • -br- (1.3.13) Under this condition g{n) = Cn+i/c„. where c„ is the n-th term of the series, is an

eUiptic function with periods a~^ and Ta~^ for q = e^^^'' and p = e^'^''^ with Im(T) > 0 and ff 0 Z + r Z .

Definition 1.3.6. A theta hypergeometric series r+iEr is called well-poised if qai = Q'i&l = . . . = ar+\br.

A theta hypergeometric series r+iEr, with r > 4, is called very-well-poised if it is well-poised and a2 = qa^ , a^ = —qa^ , 04 = qa^ p ~ ' / ^ and 05 = g a / p'^^.

Note t h a t for a well-poised theta hypergeometric series the balancing condition reduces to a\a\ • • -0^+1 = {O-IQY'^^• Since

own ^ (ga'/^ -ga'/^ qa"Vp'l\ -qa'^V; q, p)„

e{a) (0,1/2,-aV2,_ai/2/pi/2^ai/2pi/2.^^p)^ ^ 1' ^ we define the very-well-poised theta hypergeometric series r + i K by

r+iVr{auae,...ar+i:q,p;z)

^ y ^ 6 ' ( a i g 2 " ) {üuaa ar+i-q.p)n , ,n (1.3.14) ^ 9{a,) {q.aiq/ae....,aiq/ar+i:q.p)n

The series r+iK- is elliptically balanced if and only if

flfi^a^-.-a^V?" = («!'?)'•"'• (1-3-15) If 2 = 1 we suppress the argument z in the notation.

In [22] an elliptic analogue of Bailey's 101^9-transformation formula is proved. Let bcdefg = a^q^"'^'^^ and A = a^q/bcd. then

i2Vii(a;è, c.rf.e,/,,g,(?^":g,p)

(ag, g g / e / , \q/e, \q/f)n , , ,, . . . > / , ,, . _„ ^ (1.3.16) = 7—1 , , . , / . , X uVnikXb a,Xc a.M a,e.f,g.q ;q.p)

[aq/e,aq/f,\q/ef,\q)n

In Chapter 3 we prove this relation using properties of the elliptic U{2) quantum group and its representation theory.

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1.4 Overview

In this section we give an overview of the contents of this thesis.

In Chapter 2 we discuss the general concepts for dynamical quantum groups. We start with the definition of f)-Hopf algebroids introduced by Etingof and Varchenko [16], [17], which can be seen as the dynamical analogue of Hopf algebras. We give the generalized FRST-construction which defines a ()-bialgebroid from a solution of the quantum dynamical Yang-Baxter equation (1.2.4) or (1.2.5) analogous to the FRST-construction discussed in §1.2.2. We extend the definition of a pairing introduced in Roscngren [48] to dynamical quantum groups involving a spectral parameter. Using this definition of a pairing for f)-Hopf algebroids we develop the theory of actions of one f)-Hopf algebroid on another one. We also discuss the representation theory for dynamical quantiun groups, including the definition of singular and spherical vectors. For paired f)-Hopf algebroids, we can define a dynamical representation of one f)-Hüpf algebroid from a corepresentation of the other f)-Hopf algebroid. Throughout the chapter we use the dynamical GL{2) quantum group, which is the dynamical analogue of the algebra of jjolynomials on the group GL(2). as an illustrative example. Most of the results of this chapter are contained in [32].

In Chapter 3 wo study the elliptic U{2) quantum group. This example fits in the general framework developed in Chapter 2. We prove that the elliptic U{2) quantum group is self-dual in an appropriate sense. For certain corepresentations, the pair-ing of two matrix elements can be written as a terminatpair-ing balanced very-well-poised theta hypergeometric i2l/n-series, see Theorem 3.4.5. Using this result we obtain bi-orthogonality relations for these elliptic hypergeometric series, which are obtained in a different context by Frenkel and Turaev [22] and Spiridonov and Zhedanov [55], This result shows that these relations can be viewed as analogues of the orthogonality re-lations for Krawtchouk polynomials, which we obtained in §1.1 from a representation of the group SL{2,C). We also rederive the elliptic analogue of the Bailey's transfor-mation formula and the quantum dynamical Yang-Baxter equation for certain elliptic hypergeometric i2Vli-series. In [34], we obtained bi-orthogonahty relations for elliptic hypergeometric series using a dynamical representation defined in [20]. In Chapter 3 we use the theory developed in Chapter 2 to obtain these bi-orthogonality relations in a more natural way.

In Chapter 4 we study the dynamical U(n) quantum group. This dynamical quan-tum group can be seen as the dynamical analogue of the quanquan-tum U{n) group studied in [21], [43], and can also be seen as the quantum dynamical analogue of the algebra of polynomials on the group U{n) of unitary n x n-matrices. We obtain an explicit formula for the dynamical analogue of the determinant and the quantum minor deter-minants. We prove that the determinant is a group-like element that connnutes with all dynamical cjuantum minor determinants, see Lemma 4.2.1. We derive Laplace ex-pansions for the dynamical quantum minor determinants. Finally we prove that the dynamical U{n) quantum group and its coopposite are dual to each other, see Theo-rem 4.3.8. We hope to use the results of this chapter to study the dynamical analogue

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of the homogeneous space U{n)/U{n — 1) and obtain results for basic hypergeometric series from studying spherical functions, as a generahzation of the results of [43].

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Chapter 2 Dynamical quantum groups:

general concepts

This chapter contains the general algebraic concepts of the theory of dynamical (juan-tum groups. We start with the definition of the basic algebraic structures involved in the theory: f)-Hopf algebroids. The main examples follow from the generalized FRST-construction, discussed in §2.1.2. Then we introduce a kind of duality for [}-Hopf algebroids which we use to construct an action of a ()-Hopf algebroid onto another one. Finally we present the representation theory for f)-Hopf algebroids and discuss the definitions of singular and spherical vectors. For paired f)-Hopf algebroids, we define a dynamical representation of one [)-Hopf algebroid from a corepresentation of the other one. Throughout this chapter we use the dynamical U{2) quantum group as an illustrative example. In Chapter 3 we apply the theory of this chapter to the elliptic U(2) quantum group and obtain results for elliptic hypergeometric series, in Chapter 4 we study the dynamical U{n) quantum group.

2.1 Dynamical quantum groups: algebraic

struc-tures

Before starting with the definition of ()-Hopf algebroids we introduce some notation which we use in the remainder of the thesis.

Let f)* be a finite-dimensional complex vector space. The notation is influenced by a natural construction where f] occurs as the Cartan subalgebra of a semisimple Lie algebra, see [15]. This will play no role in this thesis, but we consider f) as a commutative Lie algebra with dual space f)*. Denote by Mj,. the space of meromorphic functions on f)*, and the function identically 1 on I}* by 1. For a € f)*, we denote by T„ : A/^. -^ A/„. the automorphism ( r „ / ) ( A ) = /(A + a) for all A G ()*. By Endc(^') we denote all endomorpliisms on the space A which are C-invariant, i.e. commutes with multiplication by a complex number.

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24 C H A P T E R 2. D Y N A M I C A L Q U A N T U M G R O U P S : G E N E R A L C O N C E P T S

2.1.1 Definition of J)-Hopf algebroids

D e f i n i t i o n 2 . 1 . 1 . A f)-prealgebra A is a complex vector space equipped with a decom-position A = 0 o i3g(,.-^Q/i and two left actions /x;. j^r : A/t,. —> Endc(.4), called the left and right rnom.ent map. which preserve the decomposition and such that

Mf)iir{g) = i^r{g)m{f)- for all f. g e A/t,-.

A f)-prealgebra homoinorphisin is a C-hnear map between two \)-prealgebras that preserves the decomposition and the moment maps.

Note that in case A = AQO this is just an extension of scalars to Mt^-.

D e f i n i t i o n 2 . 1 . 2 . A ()-algebra A is a \)-prealgebra which is also a unital associative algebra such that the decomposition is a bigrading for the algebra, i. e. the multiplication map m of A satisfies m : Aag x A^s —> Aa+-,,[}+6, and such that the left and right moment map fj-i. jir • A/|,. —» ^oo cire algebra embeddings by ^J-i{J) = l-ii{f)^A' iwif) = A*r(/)lyi for which the commutation relations

l'i{f)n = ani{Tcf). tir{f)a. = af^riTsf).

hold for all ƒ € A/j,- and all a G Aaj- Here 1_4 denotes the unit element of A.

A f)-algebra homomorphism is an algebra homomorphism 4> : A ^ B between two fj-algebras A and B which preserves the bigrading and the moment maps. i.e. 4>{fJ-fif)) = lifif), 0{tif{f)) = I'fif) .for all ƒ e A/^. and © ( A d ) C B,,, for all a,

pel).*

The (matrix) tensor product A(^B of two ()-algebras A and B is a [^-algebra with the foUowing definitions of the bigrading, moment maps and muhiphcation;

(AèB)^^^ = 0 ( A T ®A/,. B,0). (2.1.1a)

i^f^^{f) = i^fif)®h //;^'^«(/) = i ® / / f ( / ) , (2.1.1b) ( a ® 6)(c(g>d) = {ac) 0 {bd) for aU a. c G A. b. d£ B. (2.1.1c) where (8)A/,,, denotes the usual tensor prochict modulo the relations

fi-^{f)a ® 6 = a ® fif{f)b. for all a e A. b e B. and all ƒ G A/^.. (2.1.2)

So Aa-, is viewed as a right A/[,.-module by a • ƒ = fJ.r{f)a and B-,ij is viewed as left A/[,.-module by f • b = fii(f)b. It is straightforward to check that the multiplication (2.1.1c) is well-defined and that AèS is a f)-algebra. Indeed,

(a ® h)iîf(f)c ®d) = aîf{f)c ® bd = ii-^{T^.J)ac ®bd = ac ® fif{T^jf)bd = ac® bîf{f)d ={a® b){c ® fif{f)d).

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for a e Aa-y and b G 6-^,3. and similarly {jj.-^{f)a ®b){c®d) = {a® fj.f{f)b){c (g) d). So (2.1.1c) is well defined. Also

m^®'' : (AëBU X {A®B),s - 0 (AaaA.r ®M,. B^jBrs) CT.rel)*

Mi'®^(/)(a ® Ö) = (/*;^(/)a) ® 6 = {afif{TJ)) 0b = {a® b) ^if^" {TJ), for a ^ 6 € {A®B)a() and analogonsly for the right moment map. Note that we can define the (matrix) tensor product A®B of two ()-prealgebras A, B in the same \va\' but only requiring (2.1.1a), (2.1.1b). Then . 4 ^ 0 gives again a f)-prealgebra.

Also note that the (matrix) tensor product satisfies {A®B)®U = A®{B®U) for f)-prealgebras A. B and U.

R e m a r k 2.1.3. We use superscripts to indicate to which f)-(pre)algebra the moment maps, multiplication etc. belong if necessary. The superscripts are omitted if no confusion can arise.

E x a m p l e 2.1.4. An important example of a ()-algebra is the space ö|,. of finite

dif-ference operators "^^fiTa, on Mt,'. The bigrading is given by / T _ Q € (£*[!• )aa (so (•Cfi*)a/3 = {0} for a ^ if). The left and right moment map are equal to the natural embedding [ii{j) = /^r{f) = fTo- Let us remark that every function ƒ G M^,, can be seen as the nmltiplication operator /TQ 6

of,*-For later use we note that, as f)-prealgebras. A^D^,. ~ A ^ D^^-ISA by a ® T-0 = a = T-a ® a for a e Aap and using (2.1.2) this imphes

//;^( ƒ)«. = a ®/T__a, ^j.f{f)a = fT.^®a, for all a € A . ^ (2.1.3) This identification holds in particular for f)-algcbras. and for A = D^,. this gives the identification Dtf'^Df^. ~ D^. via / r _ „ 0gT_a = {f9)T-a- Foi' later reference we also note the identity

F G l = ( F l ) ( r _ „ G ' l ) , for a l l F e (Z?„.)na-Ge Z)„.. (2.1.4)

Definition 2.1.5. A [)-coalgebroid A is a i)-prealgehra with two \)-prealgehra mor-phisms A : ^ ^ A®A. the comultiplication. and e : A -^ op,., the counit. satisfying the coassociativity condition (A Cg) Id) o A = (Id Cg" A) o A and the counit condition (f ® Id) o A = Id = (Id (g) e) o A (using the identification (2.1.3)j.

A f)-coalgebroid homomorphism is a \)-prealgebra honiomorphism 4) : A ^ B be-tween two tj-coalgebroids A and B which preserves the comultiplication and counit, i.e. {(j) ® (l>) o A-^ = A'^ o 0 and s-^ = e'^ o 0.

E x a m p l e 2.1.6. By defining the comultiplication A : Dt,- —> £)(,.(8iD[,. ~ Dp,, by the

canonical isomorphism as defined after Example 2.1.4 and the counit e : £)(,- -^ Dt,-as the identity, we equip Df,. with the structure of a ()-coalgebroid.

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26 C H A P T E R 2. D Y N A M I C A L Q U A N T U M G R O U P S : G E N E R A L C O N C E P T S

The counit is compatible with AèDt,. c^ ^ ~ Dt,.ë>A by e{a ® fT.g) = e-^ia) i» £°''* ifT-g) arid £{fT-a ® a) = e'^"' (fT.^) 0 £-^{a) for a G Aa/s- Note that the maps £^ ® e^i* : ^(g)Z)(,. -^ D^.^Dt,. and e-^"* » e-^: D t , . ® ^ ^ L»^.®!?!,. are well-defined.

N o t a t i o n : We use Sweedler's notation for the comultiplication. i.e. A(a) = X^(a)a{i)®

0(2), where the decomposition on the right hand side is with respect to the bigrading for a homogeneous element, so a € A„i3, a(i) G ^ Q , , , a(2) & Arjg.

Define/(,. = A/[,.g)A/(,., then any ()-prealgebra ^ is a right /f,.-module by «-(/(g^) = lxi{T.af)lJ.r{T-pg)a for a G A / 3 and a left /[,^-module by (ƒ (g) 5) • a = nj{f)jir{g)a. For f)-prealgebras .4 and Ö we define A®B = ^ 0/^. Ö. Then ^ $ § 0 is again a ()-prealgebra with decomposition and moment maps given by

Aai3®B-,i C {A%B)c+-y,0+s, (2.1.5a) îf^{f){a®b)=îfU)a®b, ti^^Û){a®h) = nf{f)a®b. (2.1.5b)

If ^ and B are f)-coalgebroids, then A®B is a f)-coalgebroid with comultiplication and counit defined by

A'^®''(a (g) 6) = ^ (a(i) (g ö(i)) (g (a(2) ® 6(2)), (2.1.6a)

(a).(6)

e-^®^(« ®b) = £-^(a) o ^''(6). (2.1.6b) If moreover ^ is a f)-algobra then we can rewrite the defining relations in A ®i^, B as

cntfif) ®b = a® /if (ƒ)&, afifif) ®b=a® fif{f)b. for all a. e A. b e B. (2.1.7) E x a m p l e 2.1.7. The bigrading ƒ(,. = (/(,>)oo and moment maps //((ƒ) = f ® 1, Urif) = 1 81 ƒ equip /[,. with a f)-prealgebra structure. The comultiplication and counit defined by A ( / ® g) = / ( g l c g l i g g i and £{f 'S) g) = fg make /(,• into a ()-coalgebroid.

We can also equip /(,. with the structure of a f)-algebra by defining the multiplica-tion by (ƒ ® g){h®k) = fh® gk.

Analogously to the identification A ® Df,. ~ yl ~ Di,'®A as f}-algebras, we have A®It,' ~ .A 2i ƒ,,. ®A as f)-prealgebras by a ig 1 ig 1 = a = 1 (g 1 (g a for all a G A. So IJ.i{f)a = f ®l®a and fir{f)a = a®l® f. This identification holds in particular for f)-coalgebroids.

R e m a r k 2.1.8. The category of (i-algebras is a tensor category with product ® and

unit D(,-. The left and right unit constraint are given by / : Di,'®A -^ A and r : A®Dt,. -* A defined by l{fT_a®a) = Mf)"- and r{a®fT_g) with a € Aaff- The category of ()-coalgebroids is a tensor category with product ® and unit /(,•. The left and right unit constraints arc given by / : /p,.®A -^ A and r : A® It,- -^ A defined

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by /(ƒ (8) 5 ® a) = m{f)ti^r{9)a and r{a ® f f^ g) = fii(f)fir(9)a respectively. See e.g. [28] for the theory of tensor categories.

Let A and B be f)-bialgebroids. Then neitlier A®B nor A^B liave a natural ()-bialgebroid structure. For example A{a ^ b) = Y^u) (6)('^(i) ® ^(i)) ® (^^(2) ® ^(2)) does not define a comultiplication on A^B since A{firif)a 0 b) ^ A{a §5 /v;(/)&).

Definition 2.1.9. A fi-bialgebroid is a \:)-algebra which is also a ^-coalgebroid such that the comultiplication and the counit are \)-algebra homomorphisms.

A f)-bialgebroid homomorphism is a l)-algebra homomorphism preserving the co-multiplication and counit.

R e m a r k 2.1.10. A f}-bialgebroid A can also be defined as a f)-coalgcbroid which also has a t)-algcbra structure such that the multiplication rn : A®A -^ A and the unit 77 :ƒ(,.—> ^ defined by r]{f ® ^) = fj,i{f)iJ,,-{g) are f)-coalgebroid homomorphisms.

Definition 2.1.11. A f)-Hopf algebroid A is a \)-bialgebroid equipped with a C-linear map S: A -^ A, the antipode or co-inverse, satisfying S(iJ,r{f)a) = 5(a)/i((/) and 5(a//;(/)) = / i r ( / ) 5 ( a ) for all a e A and ƒ G M^,. and

m o ( I d ® 5 ) o A ( a ) = ;/;(s(o)l), for all a e A, m o ( 5 (S> Id) o A(a) = fij.{Tae{a.)l)). for all a e

Aas-E x a m p l e 2.1.12. Defining 5 : 0 , , . ^ P,,. by SifT„) = Tâ °f= {Tâf) o T.â. £»!,-becomes a f)-Hopf algebroid with the additional property that S is an involution.

With S : I^- —^ /(,• defined by S{f 'S) g) = 5 (S" ƒ we equip /|,. with the structure of a f)-Hopf algebroid.

R e m a r k 2 . 1 . 1 3 . (i) This definition of the antipode follows [35, §2], and it differs shghtly from [16[, see also [15]. We come back to this remark in Remark 2.1.19. (ii) It is straightforward to check that mo [SS) Id) in (2.1.8) is well-defined on AsA, and for in o (Id ® 5) we note that for a G Aa-,, b £ A^/j. ƒ 6 M^-. we have m o (Id (81 S){{f,r{f)a)^b) = iXr{f)aS{b) = aS{b)tir{f) = aS{fii{f)b) = rno{ldS)S){a^{Mm) using aS{b) G Aa-yA-p,--, C Aa-i3,Q since the first condition on 5 forces S{Aaa) ^ (iii) The antipode on a I)-Hopf algebroid is compatible with AS)D^- ~ ^ ~ P|,.(8i.4 by S{a ® fT^p) = 5^*- (fT^p) ® S^{a) and 5 ( / r _ „ » a) = S^{a) ® 5^** ( / T _ , ) using the notation of (2.1.3). Note that these maps are well-defined.

As a consequence of this definition for the antipode we have the following projio-sition due to Koelink and Rosengren [35. Prop. 2.2],

P r o p o s i t i o n 2.1.14. The antipode of a i)-Hopf algebroid A is unique and satisfies

Dynamical quantum groups: Duality and special functions

Dynamical Quantum Groups

Duality and Special Functions

Dynamical Quantum Groups

Duality and Special Functions

Preface

Contents

Bibliography 125

Summary 131

Samenvatting I33

Dankwoord I35

Curriculum Vitae I37

Chapter 1

Introduction

T T

T T

T T

1.1 T h e group 5 ^ ( 2 , C)

1.1.1 Representations of SL{2,C) and Krawtchouk

polyno-mials

E E 7 a^-'^-^b'-^c'd' a''-'-^f+'

^E"-'-"'v 2: (•::;) G)(^)'^"'->'

1.1.2 T h e Hopf algebra V(SL{2, C)) and duality

6

K ° ^ ( ; ; ) < : ^ ( w f )

There exists a pairing (•, •) : U{5l{2.C)) x P(5L(2,C)) -^ C defined by

f{exp{tX)). (1.1.6)

<^-f^=7t

\x-fM=i

figexpitX)) [f-X]{g)

Jt

1.1.3 Racah coefficients and the dynamical Yang-Bcixter

equa-tion

= Y^ Rll{NuN2.Ni + N^-2r)Jy{v2®Uv,®Jr{v3®V4)))

= L E E «pT(^i'^2,A^3 + iV4-2r)iïf,(yVi,iV3,7V4)

E E E R?qiNuN2,N3 + N,-2r)

= E E E Kri^^, N3,N,)

1,2 Quantum groups

1.2.1 Q u a n t u m Yang-Baxter equations

i?''^(A - h^'KzJz,)R''\X,z,/z3)R'\X - h^'\z,/zs)

1.2.2 FRST-construction

1.3 Special functions

1.3.1 (Basic) hypergeometric series

18

TK„,{q-^;p,N-q)K„{q-':p,N-,q)^^-^^{-pr-(-p,q '^:q)n 1+pq^"

1.3.2 Elliptic hypergeometric series

zeC\{0}. dim ar) = l[e{a,), (1.3.9)

own ^ (ga'/^ -ga'/^ qa"Vp'l\ -qa'^V; q, p)„

1.4 Overview

Chapter 2

Dynamical quantum groups:

general concepts

2.1 Dynamical quantum groups: algebraic

struc-tures

2.1.1 Definition of J)-Hopf algebroids

pel)*.

S{Aa0) C A-0,-a. S{ab) = S{b)Sia), S(Mf)) = t^rif), S{^ir(f)) = Mf),

pel).*