Elliptic quantum groups

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 Wess-Zumino-Witten model of conformal field theory on tori. More details are presented in [Fe] and complete proofs will appear in a separate publication. Quantum groups (Drinfeld-Jimbo quantum enveloping algebras, Yangians, Sklyanin algebras, see [D], [Sk]) are the algebraic structures underlying integrable models of statistical mechanics and 2-dimensional 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 interaction-round-a-face 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 Knizhnik-Zamolodchikov (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]: