Irreducible quantum group modules with finite dimensional weight spaces. I

TL;DR

This paper classifies all simple weight modules with finite dimensional weight spaces for quantum groups at roots of unity, extending classical Lie algebra results to the quantum setting.

Contribution

It provides a complete classification of simple weight modules for quantum groups at roots of unity, excluding type G2, using methods adapted from classical Lie algebra theory.

Findings

Classified all simple weight modules for quantum groups at roots of unity.

Extended classical Lie algebra classification techniques to quantum groups.

Excluded type G2 from the classification.

Abstract

In this paper we classify all simple weight modules for a quantum group $U_q$ at a complex root of unity $q$ when the Lie algebra is not of type $G_2$. By a weight module we mean a finitely generated $U_q$-module which has finite dimensional weight spaces and is a sum of those. Our approach follows the procedures used by S. Fernando and O. Mathieu to solve the corresponding problem for semisimple complex Lie algebras.