Coxeter quiver representations in fusion categories and Gabriel's theorem

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.