Semisimple and separable algebras in multi-fusion categories

TL;DR

This paper classifies semisimple and separable algebras within multi-fusion categories over arbitrary fields, extending classical algebra results and identifying conditions for separability related to field extensions and algebra dimensions.

Contribution

It provides a classification of semisimple and separable algebras in multi-fusion categories, generalizing Wedderburn-Artin theorem and linking separability to field extensions and algebra dimensions.

Findings

Separable algebras are characterized by nonvanishing dimensions.

Inseparable field extensions are the main obstructions to separability.

Division algebras are separable if and only if they have nonzero dimension.

Abstract

We give a classification of semisimple and separable algebras in a multi-fusion category over an arbitrary field in analogy to Wedderben-Artin theorem in classical algebras. It turns out that, if the multi-fusion category admits a semisimple Drinfeld center, the only obstruction to the separability of a semisimple algebra arises from inseparable field extensions as in classical algebras. Among others, we show that a division algebra is separable if and only if it has a nonvanishing dimension.