Monoidal Morita invariants for finite group algebras

TL;DR

This paper introduces monoidal Morita invariants for finite group algebras using braid group representations from the Drinfeld double, showing that the number of elements of a given order is a Morita invariant.

Contribution

It develops new invariants for finite-dimensional Hopf algebras based on braid group representations, linking algebraic properties to topological invariants.

Findings

Number of elements of order n is a monoidal Morita invariant for finite group algebras.

Established connections between these invariants and Reshetikhin-Turaev 3-manifold invariants.

Abstract

Two Hopf algebras are called monoidally Morita equivalent if module categories over them are equivalent as linear monoidal categories. We introduce monoidal Morita invariants for finite-dimensional Hopf algebras based on certain braid group representations arising from the Drinfeld double construction. As an application, we show, for any integer $n$, the number of elements of order $n$ is a monoidal Morita invariant for finite group algebras. We also describe relations between our construction and invariants of closed 3-manifolds due to Reshetikhin and Turaev.