Asymptotics of standard modules of quantum affine algebras

TL;DR

This paper studies the asymptotic behavior of standard modules in quantum affine algebras, providing explicit formulas and decompositions for their limits, especially for the case of sl_2, advancing understanding of their structure.

Contribution

It introduces a sequence of q-characters for standard modules, proves their limits as formal power series, and constructs asymptotic modules with explicit formulas for sl_2.

Findings

Established explicit limit formulas for q-characters

Constructed asymptotic standard modules for sl_2

Proved decomposition of limits into simple module q-characters

Abstract

We introduce a sequence of $q$-characters of standard modules of a quantum affine algebra and we prove it has a limit as a formal power series. For $\mathfrak{g}=\hat{\mathfrak{sl}_{2}}$, we establish an explicit formula for the limit which enables us to construct corresponding asymptotical standard modules associated to each simple module in the category $\mathcal{O}$ of a Borel subalgebra of the quantum affine algebra. Finally, we prove a decomposition formula for the limit formula into $q$-characters of simple modules in this category $\mathcal{O}$.