Finite type modules and Bethe ansatz equations

TL;DR

This paper introduces a new category of modules for quantum affine algebras with finitely many characteristic values, classifies irreducibles, and derives Bethe ansatz equations from transfer matrix operators.

Contribution

It extends the category of finite-dimensional modules, classifies irreducible objects, and establishes Bethe ansatz equations via transfer matrices and Q-operators.

Findings

Classification of irreducible modules in the new category

Identification of Baxter Q-operators and T-operators

Derivation of Bethe ansatz equations for eigenvalues

Abstract

We introduce and study a category $\text{Fin}$ of modules of the Borel subalgebra of a quantum affine algebra $U_q\mathfrak{g}$, where the commutative algebra of Drinfeld generators $h_{i,r}$, corresponding to Cartan currents, has finitely many characteristic values. This category is a natural extension of the category of finite-dimensional $U_q\mathfrak{g}$ modules. In particular, we classify the irreducible objects, discuss their properties, and describe the combinatorics of the q-characters. We study transfer matrices corresponding to modules in $\text{Fin}$. Among them we find the Baxter $Q_i$ operators and $T_i$ operators satisfying relations of the form $T_iQ_i=\prod_j Q_j+ \prod_k Q_k$. We show that these operators are polynomials of the spectral parameter after a suitable normalization. This allows us to prove the Bethe ansatz equations for the zeroes of the eigenvalues of the $Q_i$ operators acting in an arbitrary finite-dimensional representation of $U_q\mathfrak{g}$.