Asymptotic representations and Drinfeld rational fractions

TL;DR

This paper develops a new category of representations for quantum loop algebras, constructs fundamental modules as limits of known modules, and classifies simple modules via rational functions, advancing understanding of algebraic structures.

Contribution

It introduces a novel category of Borel algebra representations, constructs fundamental modules as limits, and classifies simple modules using rational functions.

Findings

Fundamental representations are constructed as limits of Kirillov-Reshetikhin modules.

Simple modules are classified by rational functions with specific regularity properties.

Explicit formulas for characters of fundamental representations are established.

Abstract

We introduce and study a category of representations of the Borel algebra, associated with a quantum loop algebra of non-twisted type. We construct fundamental representations for this category as a limit of the Kirillov-Reshetikhin modules over the quantum loop algebra and we establish explicit formulas for their characters. We prove that general simple modules in this category are classified by n-tuples of rational functions in one variable, which are regular and non-zero at the origin but may have a zero or a pole at infinity.