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Coxeter quiver representations in fusion categories and Gabriel's theorem

Edmund Heng

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.

Coxeter quiver representations in fusion categories and Gabriel's theorem

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 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 and . 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.
Paper Structure (23 sections, 18 theorems, 76 equations, 2 figures)

This paper contains 23 sections, 18 theorems, 76 equations, 2 figures.

Table of Contents

  1. Preliminaries: fusion categories and TLJ
  2. Fusion tensor categories and their module categories
  3. Temperley-Lieb-Jones category at root of unity
  4. Chebyshev polynomials and fusion ring
  5. Deligne's tensor product
  6. Coxeter quiver representations in fusion categories
  7. Coxeter quivers
  8. Representations of Coxeter quivers
  9. Unfolding and Gabriel's classification for Coxeter quivers
  10. Aside on C(Q)
  11. From Coxeter quivers to classical quivers
  12. Abelian equivalence between rep(Q) and its unfolding
  13. Gabriel's classification for Coxeter quivers
  14. Reflection functors and positive roots
  15. Root system over fusion rings and dimension vectors
  16. ...and 8 more sections

Key Result

Theorem 1

A quiver $\check{Q}$ has finitely-many isomorphism classes of indecomposable representations (finite-type) if and only if its underlying graph $\overline{\check{Q}}$ is a (finite, disjoint) union of ADE Dynkin diagrams: $A_n, D_n, E_6, E_7, E_8$. Moreover, in these cases the isomorphism classes of i

Figures (2)

  • Figure 1: Two examples of unfoldings $\check{Q}$ of Coxeter quivers $Q$, with $\check{Q}$ above $Q$.
  • Figure 2: Unfolding of finite type non-simply-laced Coxeter graphs to finite type simply-laced Coxeter graphs.

Theorems & Definitions (60)

  • Theorem : Gabriel's theorem Gab72
  • Theorem A: = \ref{['thm: gabriel classification']}
  • Theorem B: = \ref{['thm: gabriel root']}
  • Theorem C: = \ref{['thm: rep Q and checkQ equiv']}
  • definition 1.1
  • example 1.2
  • definition 1.3
  • definition 1.4
  • example 1.5
  • remark 1.6
  • ...and 50 more