Coxeter quiver representations in fusion categories and Gabriel's theorem

TL;DR

This paper extends classical quiver representation theory to Coxeter quivers using fusion categories, proving a generalized Gabriel's theorem that links indecomposable representations to Coxeter root systems.

Contribution

It introduces Coxeter quiver representations via fusion categories and proves a generalized Gabriel's theorem including non-crystallographic types.

Findings

Generalized Gabriel's theorem for Coxeter quivers.

Indecomposable representations correspond to positive roots.

Extension of classical quiver results to non-crystallographic types.

Abstract

We introduce a notion of representation for a class of generalised quivers known as Coxeter quivers. These representations are built using fusion categories associated to $U_q(\mathfrak{s}\mathfrak{l}_2)$ at roots of unity and we show that many of the classical results on representations of quivers can be generalised to this setting. Namely, we prove a generalised Gabriel's theorem for Coxeter quivers that encompasses all Coxeter--Dynkin diagrams -- including the non-crystallographic types $H$ and $I$. Moreover, a similar relation between reflection functors and Coxeter theory is used to show that the indecomposable representations correspond bijectively to the positive roots of Coxeter root systems over fusion rings.