Constructing non-semisimple modular categories with relative monoidal centers

TL;DR

This paper explores conditions under which non-semisimple modular categories can be constructed using relative monoidal centers, with applications to quantum groups and Nichols algebras, advancing the understanding of their modular properties.

Contribution

It establishes when M"uger centralizers are modular in non-semisimple categories and provides new criteria for constructing such categories via relative monoidal centers.

Findings

M"uger centralizers are modular under certain conditions

Relative monoidal centers can produce non-semisimple modular categories

Representation categories of small quantum groups are examples

Abstract

This paper is a contribution to the construction of non-semisimple modular categories. We establish when M\"uger centralizers inside non-semisimple modular categories are also modular. As a consequence, we obtain conditions under which relative monoidal centers give (non-semisimple) modular categories, and we also show that examples include representation categories of small quantum groups. We further derive conditions under which representations of more general quantum groups, braided Drinfeld doubles of Nichols algebras of diagonal type, give (non-semisimple) modular categories.