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 $\mathfrak{sl}_n$ in the same vein as the usual R–matrices gives rise to quantum groups via the RTT 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 presentation of Yangians and quantum loop algebras. I will also present a method of constructing representations of the elliptic quantum group using q–difference equations. 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.