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Elliptic quantum groups
 In Proc. XIth International Congress of Mathematical Physics
, 1995
"... This note gives an account of a construction of an “elliptic quantum group” associated with each simple classical Lie algebra. It is closely related to elliptic face models of statistical mechanics, and, in its semiclassical limit, to the WessZuminoWitten model of conformal field theory on tori. M ..."
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Cited by 102 (9 self)
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This note gives an account of a construction of an “elliptic quantum group” associated with each simple classical Lie algebra. It is closely related to elliptic face models of statistical mechanics, and, in its semiclassical limit, to the WessZuminoWitten model of conformal field theory on tori. More details are presented in [Fe] and complete proofs will appear in a separate publication. Quantum groups (DrinfeldJimbo quantum enveloping algebras, Yangians, Sklyanin algebras, see [D], [Sk]) are the algebraic structures underlying integrable models of statistical mechanics and 2dimensional conformal field theory, and found applications in several other contexts. However, from the point of view of statistical mechanics, the picture is not quite complete. In particular, elliptic interactionroundaface models of statistical mechanics have sofar escaped a description in terms of quantum groups (expect in the slN case). In this paper, we give such a description. It is hoped that the construction will shed light in other contexts, such as a description of the category of representation of quantum affine Kac–Moody algebras, or the elliptic version of Macdonald’s theory. Our definition is motivated by the following known construction that links conformal field theory to the semiclassical version of quantum groups. Conformal blocks of WZW conformal field theory on the plane obey the consistent system of KnizhnikZamolodchikov (KZ) differential equations for a function u(z1,...,zn) taking values in the tensor product of n finite dimensional representations of a simple Lie algebra g [KZ]: ∂ziu = ∑ r(zi − zj) (ij) u (1) j:j̸=i Here, the “classical rmatrix ” r(z) is the tensor C/z, where C ∈ g ⊗ g is a symmetric invariant tensor. We use the notation X (i) , for X ∈ g or End(Vi), to
The elliptic curve in the Sduality theory and Eisenstein series for KacMoody groups
"... (0.1) The goal of this paper is to develop a certain mathematical framework underlying the Sduality conjecture of Vafa and Witten [VW]. Let us recall the formulation. Let S be a smooth projective surface over C and G a semisimple algebraic group. Denote by MG(S,n) the moduli space of semistable pri ..."
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Cited by 67 (0 self)
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(0.1) The goal of this paper is to develop a certain mathematical framework underlying the Sduality conjecture of Vafa and Witten [VW]. Let us recall the formulation. Let S be a smooth projective surface over C and G a semisimple algebraic group. Denote by MG(S,n) the moduli space of semistable principal Gbundles on
Integral Representations of Solutions of the Elliptic KnizhnikZamolodchikovBernard equations
 TO APPEAR IN INTERNATIONAL MATHEMATICS RESEARCH NOTICES
, 1995
"... We give an integral representation of solutions of the elliptic Knizhnik–Zamolodchikov–Bernard equations for arbitrary simple Lie algebras. If the level is a positive integer, we obtain formulas for conformal blocks of the WZW model on a torus. The asymptotics of our solutions at critical level gi ..."
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Cited by 51 (19 self)
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We give an integral representation of solutions of the elliptic Knizhnik–Zamolodchikov–Bernard equations for arbitrary simple Lie algebras. If the level is a positive integer, we obtain formulas for conformal blocks of the WZW model on a torus. The asymptotics of our solutions at critical level gives eigenfunctions of Euler–Calogero–Moser integrable Nbody systems. As a byproduct, we obtain some remarkable integral identities involving classical theta functions.
Conformal blocks on elliptic curves and the KnizhnikZamolodchikovBernard equations
"... Abstract. We give an explicit description of the vector bundle of WZW conformal blocks on elliptic curves with marked points as subbundle of a vector bundle of Weyl group invariant vector valued theta functions on a Cartan subalgebra. We give a partly conjectural characterization of this subbundle i ..."
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Cited by 43 (9 self)
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Abstract. We give an explicit description of the vector bundle of WZW conformal blocks on elliptic curves with marked points as subbundle of a vector bundle of Weyl group invariant vector valued theta functions on a Cartan subalgebra. We give a partly conjectural characterization of this subbundle in terms of certain vanishing conditions on affine hyperplanes. In some cases, explicit calculation are possible and confirm the conjecture. The Friedan–Shenker flat connection is calculated, and it is shown that horizontal sections are solutions of Bernard’s generalization of the Knizhnik–Zamolodchikov equation. 1.
Spherical functions on affine Lie groups
 Duke Math. J
, 1995
"... hepth 9407047 ..."
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Traces of intertwiners for quantum groups and difference equations, I
 DUKE MATHEMATICAL JOURNAL
, 2000
"... ..."
Duality in Integrable Systems and Gauge Theories
, 2000
"... We discuss various dualities, relating integrable systems and show that these dualities are explained in the framework of Hamiltonian and Poisson reductions. The dualities we study shed some light on the known integrable systems as well as allow to construct new ones, double elliptic among them. We ..."
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Cited by 20 (3 self)
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We discuss various dualities, relating integrable systems and show that these dualities are explained in the framework of Hamiltonian and Poisson reductions. The dualities we study shed some light on the known integrable systems as well as allow to construct new ones, double elliptic among them. We also discuss applications to the (supersymmetric) gauge theories in various dimensions.
Modular transformations of the elliptic hypergeometric functions, Macdonald polynomials, and the shift operator
"... Abstract. We consider the space of elliptic hypergeometric functions of the sl2 type associated with elliptic curves with one marked point. This space represents conformal blocks in the sl2 WZW model of CFT. The modular group acts on this space. We give formulas for the matrices of the action in te ..."
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Cited by 8 (5 self)
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Abstract. We consider the space of elliptic hypergeometric functions of the sl2 type associated with elliptic curves with one marked point. This space represents conformal blocks in the sl2 WZW model of CFT. The modular group acts on this space. We give formulas for the matrices of the action in terms of values at roots of unity of Macdonald polynomials of the sl2 type.
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.