Elliptic quantum groups and Baxter relations

Abstract

We introduce a category $\mathcal{O}$ of modules over the elliptic quantum group of ${\mathfrak{s}\mathfrak{l}}_N$ with well-behaved $q$-character theory. We construct asymptotic modules as analytic continuation of a family of finite-dimensional modules, the Kirillov–Reshetikhin modules. In the Grothendieck ring of this category we prove two types of identities: Generalized Baxter relations in the spirit of Frenkel–Hernandez between finite-dimensional modules and asymptotic modules. Three-term Baxter TQ relations of infinite-dimensional modules.