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Classification of silted algebras for two quivers of Dynkin type $\mathbb{A}_{n}$

Zongzhen Xie, Dong Yang, Houjun Zhang

Abstract

In this paper, we give a complete classification of silted algebras for the quiver $\overrightarrow{\mathbb{A}}_{n}$ of type $\mathbb{A}_{n}$ with linear orientation and for the quiver obtained from $\overrightarrow{\mathbb{A}}_{n}$ by reversing the arrow at the unique source. Based on the classification, we also compute the number of silted algebras for these two quivers.

Classification of silted algebras for two quivers of Dynkin type $\mathbb{A}_{n}$

Abstract

In this paper, we give a complete classification of silted algebras for the quiver of type with linear orientation and for the quiver obtained from by reversing the arrow at the unique source. Based on the classification, we also compute the number of silted algebras for these two quivers.

Paper Structure

This paper contains 28 sections, 34 theorems, 62 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries on tilted and silted algebras
  3. Symmetric product
  4. Tilted algebras
  5. Silted algebras
  6. Rooted quivers with relation
  7. Full connected rooted subquivers with relation of the genealogical tree
  8. An operation
  9. Silted algebras of type $\Lambda_n$
  10. Main result
  11. Tilting modules over $\Lambda_n$
  12. Tilted algebras of type $\Lambda_n$
  13. Some subsets of ${\mathcal{A}}_t(\Lambda_n)$
  14. 2-term silting complexes of $\Lambda_n$ and silted algebras
  15. Silted algebras of type $\Gamma_{n}$
  16. ...and 13 more sections

Key Result

Lemma 2.1

Figures (3)

  • Figure 1: The Auslander--Reiten quiver of $\operatorname{\mathsf{mod}}\nolimits \Lambda_n$
  • Figure 2: The Auslander--Reiten quiver of $K^{[-1,0]}(\operatorname{\mathsf{proj}}\nolimits \Lambda_n)$
  • Figure 3: The Auslander--Reiten quiver of $K^{[-1,0]}(\operatorname{\mathsf{proj}}\nolimits \Gamma_n)$

Theorems & Definitions (62)

  • Lemma 2.1
  • proof
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6: AdachiIyamaReiten14
  • Definition 2.7
  • Theorem 2.8: BuanZhou16
  • Corollary 2.9
  • ...and 52 more