Tensor products, characters, and blocks of finite-dimensional representations of quantum affine algebras at roots of unity

TL;DR

This paper investigates the structure of finite-dimensional representations of quantum affine algebras at roots of unity, focusing on tensor products, q-characters, and block decompositions, and introduces new criteria for Weyl modules to decompose into fundamental representations.

Contribution

It extends the understanding of tensor product decompositions and block structures in the root of unity case, providing new criteria and overcoming previous limitations.

Findings

Weyl modules are tensor products of fundamental representations in the generic case.

A new sufficient condition for Weyl modules to decompose into fundamental representations at roots of unity.

Proved the condition is necessary for sl(2) case.

Abstract

We establish several results concerning tensor products, q-characters, and the block decomposition of the category of finite-dimensional representations of quantum affine algebras in the root of unity setting. In the generic case, a Weyl module is isomorphic to a tensor product of fundamental representations and this isomorphism was essential for establishing the block decomposition theorem. This is no longer true in the root of unity setting. We overcome the lack of such a tool by utilizing results on specialization of modules. Furthermore, we establish a sufficient condition for a Weyl module to be a tensor product of fundamental representations and prove that this condition is also necessary when the underlying simple Lie algebra is sl(2). We also study the braid group invariance of q-characters of fundamental representations.