Irreducible quantum group modules with finite dimensional weight spaces. II

TL;DR

This paper classifies simple quantum group modules with finite dimensional weight spaces for certain Lie algebra types, focusing on admissible highest weight modules and completing the classification for types A and C.

Contribution

It advances the classification of simple quantum group modules by reducing the problem to admissible highest weight modules and completing the classification for types A and C.

Findings

Admissible simple highest weight modules only exist for types A and C.

Complete classification of simple torsion free modules for types A and C.

Reduction of classification to admissible modules using Mathieu's procedures.

Abstract

We classify the simple quantum group modules with finite dimensional weight spaces when the quantum parameter $q$ is transcendental and the Lie algebra is not of type $G_2$. This is part $2$ of the story. The first part being Irreducible quantum group modules with finite dimensional weight spaces. I (arXiv:1504.07042). In that paper the classification is reduced to the classification of torsion free simple modules. In this paper we follow the procedures used by O. Mathieu to reduce the classification to the classification of infinite dimensional admissible simple highest weight modules. We then classify the infinite dimensional admissible simple highest weight modules and show among other things that they only exist for types $A$ and $C$. Finally we complete the classification of simple torsion free modules for types $A$ and $C$ completing the classification of the simple torsion free modules.