Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Algebras in fusion categories: species and hereditary algebras

Edmund Heng, Mateusz Stroiński

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.

Algebras in fusion categories: species and hereditary algebras

Abstract

We initiate the study of non-semisimple algebras in fusion categories by establishing the framework of -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 -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.
Paper Structure (18 sections, 71 theorems, 47 equations)

This paper contains 18 sections, 71 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Abelian categories
  4. Tensor and fusion categories
  5. Module categories
  6. Module categories are categories of modules
  7. Bimodules, ideals and the C-radical of an algebra
  8. Tensor algebras
  9. Division algebras
  10. Artin--Wedderburn theorem
  11. Species, quivers and path algebras
  12. The fusion category setting
  13. Radicals of algebras in fusion categories and semisimplicity
  14. Basic semisimples and projective generators
  15. Internal Ext and directed graphs of module categories
  16. ...and 3 more sections

Key Result

Theorem 1

Let $A \in \mathcal{C}$ be an algebra in a separable fusion category $\mathcal{C}$. There exists a $\mathcal{C}$-species Q and a two-sided ideal $I$ in its path algebra $T_Q$ such that $A$ is Morita equivalent to the algebra $T_Q/I$. Moreover, $I$ is trivial if and only if $A$ is hereditary, i.e. $\

Theorems & Definitions (198)

  • Theorem 1: = \ref{['extspecies']} and \ref{['noidealpaths']}
  • Remark 1
  • Theorem 2: \ref{['quiversviaGr']} and \ref{['cor:fusionquivercondition']}
  • Remark 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Definition 8
  • ...and 188 more