Globally Irreducible Weyl Modules for Quantum Groups

TL;DR

This paper establishes criteria for the irreducibility of Weyl modules over quantum groups at roots of unity, extending classical results for algebraic groups to the quantum setting.

Contribution

It provides an analogous criterion for irreducibility of Weyl modules over quantum groups, generalizing known results from algebraic groups.

Findings

Irreducibility characterized by isomorphism to the adjoint representation or minuscule highest weight

Criteria applicable for quantum groups at roots of unity

Extends classical irreducibility conditions to quantum algebra context

Abstract

The authors proved that a Weyl module for a simple algebraic group is irreducible over every field if and only if the module is isomorphic to the adjoint representation for $E_{8}$ or its highest weight is minuscule. In this paper, we prove an analogous criteria for irreducibility of Weyl modules over the quantum group $U_{\zeta}({\mathfrak g})$ where ${\mathfrak g}$ is a complex simple Lie algebra and $\zeta$ ranges over roots of unity.