Algebraic structures in group-theoretical fusion categories

TL;DR

This paper generalizes the classification of indecomposable, separable algebras in group-theoretical fusion categories by constructing explicit Morita equivalence class representatives using a functorial approach involving twisted group algebras.

Contribution

It introduces a new method to explicitly construct Morita equivalence class representatives of indecomposable, separable algebras in group-theoretical fusion categories via a monoidal functor and twisted group algebras.

Findings

Constructed explicit Morita class representatives in group-theoretical fusion categories.

Showed twisted group algebras admit Frobenius algebra structures in pointed fusion categories.

Demonstrated the resulting algebras have desirable algebraic properties.

Abstract

It was shown by Ostrik (2003) and Natale (2017) that a collection of twisted group algebras in a pointed fusion category serve as explicit Morita equivalence class representatives of indecomposable, separable algebras in such categories. We generalize this result by constructing explicit Morita equivalence class representatives of indecomposable, separable algebras in group-theoretical fusion categories. This is achieved by providing the free functor $\Phi$ from fusion category to a category of bimodules in the original category with a (Frobenius) monoidal structure. Our algebras of interest are then constructed as the image of twisted group algebras under $\Phi$. We also show that twisted group algebras admit the structure of Frobenius algebras in a pointed fusion category, and as a consequence, our algebras are Frobenius algebras in a group-theoretical fusion category. They also enjoy several good algebraic properties.