# Autoequivalences of tensor categories attached to quantum groups at roots of $1$

## TL;DR

This paper calculates the group of braided tensor autoequivalences and the Brauer-Picard group for the representation category of small quantum groups at roots of unity, advancing understanding of their symmetries.

## Contribution

It provides explicit computations of autoequivalence groups for tensor categories associated with quantum groups at roots of unity, a previously uncharted area.

## Key findings

- Determined the group of braided tensor autoequivalences.
- Computed the Brauer-Picard group of the category.
- Enhanced understanding of symmetries in quantum group categories.

## Abstract

We compute the group of braided tensor autoequivalences and the Brauer-Picard group of the representation category of the small quantum group $\mathfrak{u}_q(\mathfrak{g})$, where $q$ is a root of unity.