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On maximal green sequence for quivers arising from weighted projective lines

Changjian Fu, Shengfei Geng

Abstract

We investigate the existence and non-existence of maximal green sequences for quivers arising from weighted projective lines. Let $Q$ be the Gabreil quiver of the endomorphism algebra of a basic cluster-tilting object in the cluster category $\mathcal{C}_\mathbb{X}$ of a weighted projective line $\mathbb{X}$. It is proved that there exists a quiver $Q'$ in the mutation equivalence class $\operatorname{Mut}(Q)$ such that $Q'$ admits a maximal green sequence. On the other hand, there is a quiver in $\operatorname{Mut}(Q)$ which does not admit a maximal green sequence if and only if $\mathbb{X}$ is of wild type.

On maximal green sequence for quivers arising from weighted projective lines

Abstract

We investigate the existence and non-existence of maximal green sequences for quivers arising from weighted projective lines. Let be the Gabreil quiver of the endomorphism algebra of a basic cluster-tilting object in the cluster category of a weighted projective line . It is proved that there exists a quiver in the mutation equivalence class such that admits a maximal green sequence. On the other hand, there is a quiver in which does not admit a maximal green sequence if and only if is of wild type.

Paper Structure

This paper contains 12 sections, 17 theorems, 35 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quivers and mutation
  4. Maximal green sequence
  5. Tropical dualities between $c$-vectors and $g$-vectors
  6. Finite mutation type
  7. Quivers arising from Weighted projective lines
  8. Weighted projective lines
  9. The category $\operatorname{\mathsf{coh}}\mathbb{X}$ of coherent sheaves
  10. Quivers associated with $T_{\text{can}}(\mathbb{X})$ and $T_{\text{sq}}(\mathbb{X})$
  11. The classification
  12. The existence and non-existence of maximal green sequence

Key Result

Theorem 2.4

DerksenWeymanZelevinsky Every mutable vertex of $R\in \mathop{\mathrm{Mut}}\nolimits(\widehat{Q},Q_0^\ast)$ is either green or red.

Figures (10)

  • Figure 2.1: Quivers of tubular type $D_4^{(1,1)}$, $E_6^{(1,1)}$, $E_7^{(1,1)}$ and $E_8^{(1,1)}$.
  • Figure 3.2: Quiver $Q_{T_{\text{sq}}(\mathbb{X})}$ with weight sequence $(p_1, \dots, p_t)$.
  • Figure 3.3: Quiver $Q_{T_{\text{can}}(\mathbb{X})}$ with weight sequence $(p_1,\dots, p_t)$, where the label $t-2$ means that there are $t-2$ arrows from $\mathcal{O}(\vec{c})$ to $\mathcal{O}.$
  • Figure 4.4: Quiver $Q_{t}=Q_{T_{\text{sq}}(\mathbb{X})}$ with weight sequence $(2,2,\dots, 2)$.
  • Figure 4.5: Quiver $Q(C)$.
  • ...and 5 more figures

Theorems & Definitions (33)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Remark 2.5
  • Definition 2.6
  • Proposition 2.7
  • Remark 2.8
  • Example 2.9
  • Lemma 2.10
  • ...and 23 more