On small modules for quantum groups at roots of unity

TL;DR

This paper investigates the minimal dimensions of irreducible modules for quantum groups at roots of unity, examining conjectures and the existence of modules meeting certain bounds.

Contribution

It analyzes the conjecture by De Concini, Kac, and Procesi, showing that modules attaining the minimal dimension bound do not always exist.

Findings

The minimal dimension bound cannot always be achieved.

The paper discusses variants and limitations of the conjecture.

Provides insights into the structure of modules at roots of unity.

Abstract

A conjecture of De Concini Kac and Procesi provides a bound on the minimal possible dimension of an irreducible module for quantized enveloping algebras at an odd root of unity. We pose the problem of the existence of modules whose dimension equals this bound. We show that this question cannot have a positive answer in full generality and discuss variants of this question.