The trace on projective representations of quantum groups

TL;DR

This paper investigates the structure of categories of weight modules over specific quantum groups at roots of unity, establishing the existence of a modified trace and ribbon structure in these categories.

Contribution

It proves the existence of a modified trace on projective modules for small and un-restricted quantum groups and shows that the unrolled quantum group category is ribbon.

Findings

Existence of a modified trace on projective modules

Category over unrolled quantum group is ribbon

Partial prior results are extended and completed

Abstract

For certain roots of unity, we consider the categories of weight modules over three quantum groups: small, un-restricted and unrolled. The first main theorem of this paper is to show that there is a modified trace on the projective modules of the first two categories. The second main theorem is to show that category over the unrolled quantum group is ribbon. Partial results related to these theorems were known previously.