# On semisimple quasitriangular Hopf algebras of dimension $dq^n$

## TL;DR

This paper proves that semisimple quasitriangular Hopf algebras with dimensions of the form $dq^n$ are solvable, and classifies those with small exponents as either abelian group algebras or twist equivalents of certain extensions.

## Contribution

It establishes solvability for a broad class of semisimple quasitriangular Hopf algebras and classifies those with small exponents, extending understanding of their structure.

## Key findings

- All such Hopf algebras are solvable.
- For $n \\leq 3$, they are either abelian group algebras or twist equivalent to specific extensions.

## Abstract

Let $q>2$ be a prime number, $d$ be an odd square-free natural number, and $n$ be a non-negative integer. We prove that a semisimple quasitriangular Hopf algebra of dimension $dq^n$ is solvable in the sense of Etingof, Nikshych and Ostrik. In particular, if $n\leq 3$ then it is either isomorphic to $k^G$ for some abelian group $G$, or twist equivalent to a Hopf algebra which fits into a cocentral abelian exact sequence.