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15
Exchange dynamical quantum groups
 Comm. Math. Phys
, 1999
"... Abstract. For any simple Lie algebra g and any complex number q which is not zero or a nontrivial root of unity, we construct a dynamical quantum group (Hopf algebroid), whose representation theory is essentially the same as the representation theory of the quantum group Uq(g). This dynamical quantu ..."
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Cited by 43 (9 self)
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Abstract. For any simple Lie algebra g and any complex number q which is not zero or a nontrivial root of unity, we construct a dynamical quantum group (Hopf algebroid), whose representation theory is essentially the same as the representation theory of the quantum group Uq(g). This dynamical quantum group is obtained from the fusion and exchange relations between intertwining operators in representation theory of Uq(g), and is an algebraic structure standing behind these relations. 1.
Spherical functions on affine Lie groups
 Duke Math. J
, 1995
"... hepth 9407047 ..."
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Hypergeometric solutions of trigonometric KZ equations satisfy dynamical difference equations
 Adv. Math
"... {yavmar, anv} @ email.unc.edu Abstract. The trigonometric KZ equations associated to a Lie algebra g depend on a parameter λ ∈ h where h ⊂ g is a Cartan subalgebra. A system of dynamical difference equations with respect to λ compatible with the KZ equations is introduced in [TV]. We prove that the ..."
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Cited by 16 (8 self)
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{yavmar, anv} @ email.unc.edu Abstract. The trigonometric KZ equations associated to a Lie algebra g depend on a parameter λ ∈ h where h ⊂ g is a Cartan subalgebra. A system of dynamical difference equations with respect to λ compatible with the KZ equations is introduced in [TV]. We prove that the standard hypergeometric solutions of the trigonometric KZ equations associated to slN also satisfy the dynamical difference equations.
Algebraic Integrability Of Macdonald Operators And Representations Of Quantum Groups
"... In this paper we construct examples of commutative rings of difference operators with matrix coefficients from representation theory of quantum groups, generalizing the results of our previous paper [ES] to the qdeformed case. A generalized BakerAkhiezer function \Psi is realized as a matrix ch ..."
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Cited by 13 (3 self)
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In this paper we construct examples of commutative rings of difference operators with matrix coefficients from representation theory of quantum groups, generalizing the results of our previous paper [ES] to the qdeformed case. A generalized BakerAkhiezer function \Psi is realized as a matrix character of a Verma module and is a common eigenfunction for a commutative ring of difference operators.
Multicomponent Calogero model of BNtype confined in harmonic potential Phys
 Lett
, 1995
"... A new onedimensional model, the multicomponent Calogero model of BNtype confined in the harmonic potential, is introduced. The Lax pair of this model is determined, and then a set of functionally independent conserved operators are constructed. Moreover, the energy spectrums of the above model are ..."
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Cited by 13 (0 self)
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A new onedimensional model, the multicomponent Calogero model of BNtype confined in the harmonic potential, is introduced. The Lax pair of this model is determined, and then a set of functionally independent conserved operators are constructed. Moreover, the energy spectrums of the above model are obtained by three different methods. Typeset using REVTEX
RepresentationTheoretic Proof Of The Inner Product And Symmetry Identities For Macdonald's Polynomials
 Compositio Math
, 1996
"... This paper is a continuation of our papers [EK1, EK2]. In [EK2] we showed that for the root system An\Gamma1 one can obtain Macdonald's polynomials  a new interesting class of symmetric functions recently defined by I. Macdonald [M1]  as weighted traces of intertwining operators between cer ..."
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Cited by 12 (1 self)
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This paper is a continuation of our papers [EK1, EK2]. In [EK2] we showed that for the root system An\Gamma1 one can obtain Macdonald's polynomials  a new interesting class of symmetric functions recently defined by I. Macdonald [M1]  as weighted traces of intertwining operators between certain finitedimensional representations of U q sl n . The main goal of the present paper is to use this construction to give a representationtheoretic proof of Macdonald's inner product and symmetry identities for the root system An\Gamma1 . Macdonald's inner product identities (see [M2]) have been proved by combinatorial methods by Macdonald (unpublished) for the root system An\Gamma1 and by Cherednik in the general case; symmetry identities for the root system An\Gamma1 have also been proved by Macdonald ([Macdonald, private communication]). The paper is organized as follows. In Section 1 we briefly list the basic definitions. In Section 2 we define Macdonald's polynomials P and recall the construction of
Central Elements For Quantum Affine Algebras And Affine Macdonald's Operators
"... We describe a generalization of Drinfeld's description of the center of a quantum group to the case of quantum affine algebras. We use the obtained central elements to construct the affine analogue of Macdonald's difference operators. ..."
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Cited by 7 (2 self)
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We describe a generalization of Drinfeld's description of the center of a quantum group to the case of quantum affine algebras. We use the obtained central elements to construct the affine analogue of Macdonald's difference operators.
Cental elements for quantum affine . . .
, 1994
"... We describe a generalization of Drinfeld’s description of the center of a quantum group to the case of quantum affine algebras. We use the obtained central elements to construct the affine analogue of Macdonald’s difference operators. ..."
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We describe a generalization of Drinfeld’s description of the center of a quantum group to the case of quantum affine algebras. We use the obtained central elements to construct the affine analogue of Macdonald’s difference operators.
The Fine Structure Of Translation Functors
, 1999
"... Let E be a simple finitedimensional representation of a semisimple Lie algebra with extremal weight # and let 0 #= e # E# . Let M(#) be the Verma module with highest weight # and 0 #= v# # M(#)# . We investigate the projection of e# v# # E# M(#) on the central character #(# + #). This is ..."
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Let E be a simple finitedimensional representation of a semisimple Lie algebra with extremal weight # and let 0 #= e # E# . Let M(#) be the Verma module with highest weight # and 0 #= v# # M(#)# . We investigate the projection of e# v# # E# M(#) on the central character #(# + #). This is a rational function in # and we calculate its poles and zeros. We then apply this result in order to compare translation functors.