Algebras in fusion categories: species and hereditary algebras

TL;DR

This paper develops a framework for studying non-semisimple algebras within fusion categories, generalizing classical algebra results and establishing conditions for hereditary algebras.

Contribution

It introduces the concept of $ ext{C}$-species in fusion categories and characterizes hereditary algebras via Morita equivalence to path algebra quotients.

Findings

Any algebra is Morita equivalent to an admissible quotient of a $ ext{C}$-species path algebra.

Hereditary algebras correspond to cases with no further quotients needed.

Results generalize Gabriel's theorem to fusion categories.

Abstract

We initiate the study of non-semisimple algebras in fusion categories by establishing the framework of $\mathcal{C}$-species -- analogous to the framework of species and quivers used in the study of Artin algebras. Under the (necessary) assumption that the fusion category is separable, we show that any algebra is Morita equivalent to an admissible quotient of the path algebra of a $\mathcal{C}$-species. Moreover, we show that an algebra is hereditary if and only if no further quotient is required. These results generalise that of Gabriel's for finite-dimensional algebras.