Classification of finite type fusion quivers

TL;DR

This paper introduces fusion quivers, generalizes Coxeter quivers, classifies finite type fusion quivers, and establishes a generalized quantum McKay correspondence within fusion categories, extending classical representation theory results.

Contribution

It defines fusion quivers, classifies those of finite type, and proves an analogue of Gabriel's theorem and the quantum McKay correspondence.

Findings

Finite type fusion quivers are classified.

An analogue of Gabriel's theorem is established.

A generalized quantum McKay correspondence is proved.

Abstract

In recent work, the second author introduced the concept of Coxeter quivers, generalizing several previous notions of a quiver representation. Finite type Coxeter quivers were classified, and their indecomposable objects were shown to be in bijection with positive roots, generalizing a classical theorem of Gabriel. In this paper we define fusion quivers, a natural generalization of Coxeter quivers. We classify the finite type fusion quivers, and prove the analogue of Gabriel's theorem. As a special case, this proves a generalised quantum McKay correspondence for fusion categories, an analogue of Auslander--Reiten's result for finite groups in the fusion categorical setting.