Verma and simple modules for quantum groups at non-abelian groups

TL;DR

This paper introduces Verma modules for quantum groups associated with non-abelian groups, establishing their structure and linking simple modules over these quantum groups to those over the Drinfeld double of the group.

Contribution

It defines Verma modules in this setting and proves their simple head and socle, establishing a correspondence between simple modules of the quantum group and the Drinfeld double.

Findings

Verma modules have simple head and socle.

Bijective correspondence between simple modules of quantum group and Drinfeld double.

Explicit description of modules for the symmetric group S3 case.

Abstract

The Drinfeld double D of the bosonization of a finite-dimensional Nichols algebra B(V) over a finite non-abelian group G is called a quantum group at a non-abelian group. We introduce Verma modules over such a quantum group D and prove that a Verma module has simple head and simple socle. This provides two bijective correspondences between the set of simple modules over D and the set of simple modules over the Drinfeld double D(G). As an example, we describe the lattice of submodules of the Verma modules over the quantum group at the symmetric group S3 attached to the 12-dimensional Fomin-Kirillov algebra, computing all the simple modules and calculating their dimensions.