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Actions of Fusion Categories on Path Algebras

Alexander Betz

TL;DR

The paper develops a framework for based actions of fusion categories on algebras, encoding actions as monoidal functors into bimodule categories and organizing them into a 2-category. It shows that separable based actions on path algebras are governed by semisimple C-module categories and dual endofunctors in End_C(M), generalizing graded/filtered Hopf-action results. A key result is that for the PSU(2)_{p-2} family, all separable idempotent split based actions on path algebras are graded, classified by partitions of vertices and fusion-coefficient data, with a thorough analysis of eigenstructure and S-matrix properties guiding the classification. This provides a complete, finitary description of quantum symmetries on path algebras in this setting and extends prior tensor-algebra classifications to a broader fusion-category context. The work has implications for understanding categorical symmetries in algebraic structures and offers a concrete, computable conduit between module categories, endofunctors, and quiver data.

Abstract

In this article, we introduce the notion of a based action of a fusion category on an algebra. We will build some general theory to motivate our interest in based actions, and then apply this theory to understand based actions of fusion categories on path algebras. Our results demonstrate that a separable idempotent split based action of a fusion category $C$ on a path algebra can be characterized in terms of $C$ module categories and their associated module endofunctors. As a specific application, we fully classify separable idempotent split based actions of the family of fusion categories $\text{PSU}(2)_{p-2}$ on path algebras up to conjugacy.

Actions of Fusion Categories on Path Algebras

TL;DR

The paper develops a framework for based actions of fusion categories on algebras, encoding actions as monoidal functors into bimodule categories and organizing them into a 2-category. It shows that separable based actions on path algebras are governed by semisimple C-module categories and dual endofunctors in End_C(M), generalizing graded/filtered Hopf-action results. A key result is that for the PSU(2)_{p-2} family, all separable idempotent split based actions on path algebras are graded, classified by partitions of vertices and fusion-coefficient data, with a thorough analysis of eigenstructure and S-matrix properties guiding the classification. This provides a complete, finitary description of quantum symmetries on path algebras in this setting and extends prior tensor-algebra classifications to a broader fusion-category context. The work has implications for understanding categorical symmetries in algebraic structures and offers a concrete, computable conduit between module categories, endofunctors, and quiver data.

Abstract

In this article, we introduce the notion of a based action of a fusion category on an algebra. We will build some general theory to motivate our interest in based actions, and then apply this theory to understand based actions of fusion categories on path algebras. Our results demonstrate that a separable idempotent split based action of a fusion category on a path algebra can be characterized in terms of module categories and their associated module endofunctors. As a specific application, we fully classify separable idempotent split based actions of the family of fusion categories on path algebras up to conjugacy.

Paper Structure

This paper contains 11 sections, 59 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quivers and Path Algebras
  4. Tensor Categories
  5. Algebra Objects and Module Categories
  6. The Two Category of Based Algebra Actions
  7. Endofunctor Subcategories
  8. The Based Bimodules of Path Algebras and Endofuntors
  9. Based Actions of Fusion Categories on Path Algebras
  10. Graded Actions
  11. Based Actions of the PSU(2) on Path Algebras

Key Result

Theorem 1.1

There is an equivalence of monoidal categories between $\operatorname{BasedBim}(\operatorname{\Bbbk Q})_{I_v}$ and $\operatorname{End}_{(\widetilde{Q},m,i)}(\operatorname{Vec}(M))$, where $M$ is the semisimple category equivalent to $\operatorname{Mod}(\operatorname{\Bbbk Q}_0)$.

Theorems & Definitions (154)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1: Path Algebra
  • Definition 2.2: The quiver generated by $\widetilde{Q}$
  • Remark 1
  • Example 2.3
  • Definition 2.4
  • ...and 144 more