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Maximal Rigid Representations of Continuous Quivers of Type A with Automorphism

Xiaowen Gao, Minghui Zhao

Abstract

Buan and Krause gave a classification of maximal rigid representations for cyclic quivers and counted the number of isomorphism classes. By using this result, we give a formula on the number of isomorphism classes of a kind of maximal rigid representations for continuous quivers of type A with automorphism.

Maximal Rigid Representations of Continuous Quivers of Type A with Automorphism

Abstract

Buan and Krause gave a classification of maximal rigid representations for cyclic quivers and counted the number of isomorphism classes. By using this result, we give a formula on the number of isomorphism classes of a kind of maximal rigid representations for continuous quivers of type A with automorphism.

Paper Structure

This paper contains 11 sections, 8 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Continuous quivers of type A with automorphism
  3. Continuous quivers of type $A$ with automorphism
  4. Maximal rigid representations
  5. Representations of type $\alpha$
  6. Titling representations for cyclic quivers
  7. Quivers and representations
  8. Representations for cyclic quivers
  9. Representations of quiver of type $A_{\infty}$ with automorphism
  10. Proof of Theorem \ref{['main result']}
  11. Continuous quivers of type $\tilde{A}$

Key Result

Lemma 2.1

Let $\mathbf{M}$ be an object of category $\mathrm{Rep}'_{k}(\mathbf{Q})$. Then the represntation $\mathbf{M}$ is rigid in $\mathrm{Rep}'_{k}(\mathbf{Q})$ if and only if it is rigid in $\mathrm{Rep}'_{k}(A_\mathbb{R})$.

Theorems & Definitions (15)

  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Example 2.3
  • Theorem 2.4
  • Proposition 3.1: 2004Tilting
  • Lemma 3.2
  • proof
  • Corollary 3.3
  • ...and 5 more