Baxter operators and asymptotic representations

TL;DR

This paper develops a category of representations for elliptic quantum groups related to rak{sl}_2, introduces Baxter Q-operators as transfer matrices, and derives key relations in the Grothendieck ring.

Contribution

It constructs a well-behaved representation category and establishes the connection between Baxter operators and asymptotic representations within this framework.

Findings

Derived separation of variables relations for asymptotic representations.

Constructed Baxter Q-operators as transfer matrices.

Established TQ-relations from Grothendieck ring relations.

Abstract

We introduce a category $\mathcal O$ of representations of the elliptic quantum group associated with $\mathfrak{sl}_2$ with well-behaved $q$-character theory. We derive separation of variables relations for asymptotic representations in the Grothendieck ring of this category. Baxter $Q$-operators are obtained as transfer matrices for asymptotic representations and obey $TQ$-relations as a consequence of the relations in $K_0(\mathcal O)$.