Three natural subgroups of the Brauer-Picard group of a Hopf algebra with applications

TL;DR

This paper constructs three natural subgroups of the Brauer-Picard group for categories of representations of finite-dimensional Hopf algebras, revealing a decomposition similar to Bruhat decomposition and exploring applications in quantum groups and topological field theories.

Contribution

It introduces explicit constructions of three subgroups of the Brauer-Picard group, providing new insights into their structure and applications in quantum algebra and topological physics.

Findings

Decomposition of the Brauer-Picard group into three subgroups

Construction of three types of braided autoequivalences

Applications to quantum groups and Nichols algebras

Abstract

In this article we construct three explicit natural subgroups of the Brauer-Picard group of the category of representations of a finite-dimensional Hopf algebra. In examples the Brauer Picard group decomposes into an ordered product of these subgroups, somewhat similar to a Bruhat decomposition. Our construction returns for any Hopf algebra three types of braided autoequivalences and correspondingly three families of invertible bimodule categories. This gives examples of so-called (2-)Morita equivalences and defects in topological field theories. We have a closer look at the case of quantum groups and Nichols algebras and give interesting applications. Finally, we briefly discuss the three families of group-theoretic extensions.