# Elliptic quantum groups and Baxter relations

## TL;DR

This paper develops a new category of modules over elliptic quantum groups, constructs asymptotic modules via analytic continuation, and proves generalized Baxter relations and TQ relations within this framework.

## Contribution

It introduces a category O for elliptic quantum groups, constructs asymptotic modules, and establishes new Baxter relations and TQ relations in this setting.

## Key findings

- Established generalized Baxter relations between finite-dimensional and asymptotic modules.
- Proved three-term Baxter TQ relations for infinite-dimensional modules.
- Constructed asymptotic modules as analytic continuations of Kirillov--Reshetikhin modules.

## Abstract

We introduce a category O of modules over the elliptic quantum group of sl_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.