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Gabriel's Theorem for Locally Finite-Dimensional Representations of Infinite Quivers

Nathaniel Gallup, Stephen Sawin

Abstract

We prove a version of Gabriel's theorem for locally finite-dimensional representations of infinite quivers. Specifically, we show that if $Ω$ is any connected quiver, the category of locally finite-dimensional representations of $Ω$ has unique representation type (meaning no two indecomposable representations have the same dimension vector) if and only if the underlying graph of $Ω$ is a generalized ADE Dynkin diagram (i.e. one of $A_n, D_n, E_6, E_7, E_8, A_{\infty}, A_{\infty , \infty}$ or $D_\infty$). This result is companion to earlier work of the authors generalizing Gabriel's theorem to infinite quivers with different conditions.

Gabriel's Theorem for Locally Finite-Dimensional Representations of Infinite Quivers

Abstract

We prove a version of Gabriel's theorem for locally finite-dimensional representations of infinite quivers. Specifically, we show that if is any connected quiver, the category of locally finite-dimensional representations of has unique representation type (meaning no two indecomposable representations have the same dimension vector) if and only if the underlying graph of is a generalized ADE Dynkin diagram (i.e. one of or ). This result is companion to earlier work of the authors generalizing Gabriel's theorem to infinite quivers with different conditions.

Paper Structure

This paper contains 10 sections, 21 theorems, 5 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Quivers
  4. Representations of Quivers
  5. The Transport of a Subspace
  6. Poset Filtrations and Almost Gradations
  7. Decompositions of Locally Finite-dimensional Representations of $A_{\infty, \infty}$ Quivers
  8. Locally Finite-dimensional Indecomposables of Finitely Branching Tree Quivers
  9. Infinite Gabriel's Theorem for Locally Finite-Dimensional Representations
  10. Future Work

Key Result

Theorem A

Let $\Omega$ be a connected quiver. The category $\operatorname{rep}(\Omega)$ has unique representation type if and only if $\Omega$ is a generalized ADE Dynkin quiver (see Figure fig: generalized ade quivers) and in this case, taking dimension vectors gives a bijection between the set of isomorphis

Figures (1)

  • Figure 1: The Generalized ADE Dynkin Quivers, where $m = 6, 7 , 8$

Theorems & Definitions (28)

  • Theorem A: Locally Finite-Dimensional Infinite Gabriel's Thm.
  • Proposition 2.1
  • Proposition 3.1
  • Lemma 3.1
  • Corollary 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Lemma 3.2
  • Lemma 3.3
  • Proposition 3.4
  • ...and 18 more