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Affine root systems, stable tubes and a conjecture by Geiss-Leclerc-Schröer

Zengqiang Lin, Xiuping Su

Abstract

Associated to a symmetrisable Cartan matrix $C$, Geiss-Lerclerc-Schröer constructed and studied a class of Iwanaga-Gorenstein algebras $H$. They proved a generalised version of Gabriel's Theorem, that is, the rank vectors of $τ$-locally free $H$-modules are the positive roots of type $C$ when $C$ is of finite type, and conjectured that this is true for any $C$. In this paper, we look into this conjecture when $C$ is of affine type. We construct explicitly stable tubes, some of which have rigid mouth modules, while others not. We deduce that any positive root of type $C$ is the rank vector of some $τ$-locally free $H$-module. However, the converse is not true in general. Our construction shows that there are $τ$-locally free $H$-modules whose rank vectors are not roots, when $C$ is of type $\widetilde{\mathbb{B}}_n$, $\widetilde{\mathbb{CD}}_n$, $\widetilde{\mathbb{F}}_{41}$ and $\widetilde{\mathbb{G}}_{21}$, and so the conjecture fails in these four types.

Affine root systems, stable tubes and a conjecture by Geiss-Leclerc-Schröer

Abstract

Associated to a symmetrisable Cartan matrix , Geiss-Lerclerc-Schröer constructed and studied a class of Iwanaga-Gorenstein algebras . They proved a generalised version of Gabriel's Theorem, that is, the rank vectors of -locally free -modules are the positive roots of type when is of finite type, and conjectured that this is true for any . In this paper, we look into this conjecture when is of affine type. We construct explicitly stable tubes, some of which have rigid mouth modules, while others not. We deduce that any positive root of type is the rank vector of some -locally free -module. However, the converse is not true in general. Our construction shows that there are -locally free -modules whose rank vectors are not roots, when is of type , , and , and so the conjecture fails in these four types.
Paper Structure (32 sections, 51 theorems, 105 equations)

This paper contains 32 sections, 51 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. The Iwanaga-Gorenstein algebras $H$ and $\tau$-locally free $H$-modules
  3. The algebra $H(C,D,\Omega)$
  4. $\tau$-locally free modules
  5. The root system $\Delta$ of type $C$
  6. Coxeter transformations, reflection functors and AR-translation
  7. Rank vectors of $\tau$-locally free $H$-modules and positive roots
  8. Regular components of AR-quivers
  9. More on $\tau$-locally free modules
  10. Conjecture \ref{['Conj:gls']} and the orientation $\Omega$
  11. Conjecture \ref{['Conj:gls']}$\colon$the forward implication
  12. Homogeneous tubes of type $\operatorname{\widetilde{\mathbb{A}}}_{11}, \operatorname{\widetilde{\mathbb{A}}}_{12}$, $\operatorname{\widetilde{\mathbb{B}}}_{n}$ and $\operatorname{\widetilde{\mathbb{G}}}_{21}$
  13. Type $\operatorname{\widetilde{\mathbb{A}}}_{11}$
  14. Type $\operatorname{\widetilde{\mathbb{A}}}_{12}$
  15. Type $\operatorname{\widetilde{\mathbb{B}}}_{n}$
  16. ...and 17 more sections

Key Result

Theorem 2

Let $C$ be a symmetrisable Cartan matrix of affine type and $H=H(C,D,\Omega)$. Then for any positive root $\alpha$ of type $C$, there exists a $\tau$-locally free $H$-module $M$ such that $\operatorname{\underline{rank}} M=\alpha$.

Theorems & Definitions (77)

  • Conjecture 1
  • Theorem 2: Theorem \ref{['thm3.1']}
  • Theorem 3: Theorem \ref{['main2']}
  • Proposition 2.1
  • Proposition 2.2
  • Example 2.3
  • Lemma 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Proposition 2.7
  • ...and 67 more