Elliptic Quantum Groups
In 1995 G. Felder introduced an elliptic R–matrix, which quantizes the classical dynamical r–matrix arising from the study of conformal blocks on elliptic curves. The elliptic R–matrix satisfies a dynamical analog of the Yang–Baxter equation and can be used to define the elliptic quantum group of sln in the same vein as the usual R–matrices gives rise to quantum groups via the RT T formalism of Faddeev, Reshetikhin and Takhtajan.
In this talk I will explain Felder’s definition and present its generalization to the case of arbitrary Kac–Moody Lie algebras analogous to the Drinfeld’s new pre- sentation of Yangians and quantum loop algebras. I will also present a method of constructing representations of the elliptic quantum group using q–difference equa- tions. Our construction gives rise to a classification of irreducible representations of the elliptic quantum group, which is reminiscent of the Drinfeld’s classification of irreducible representations of Yangians.
