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Iterated mutations of symmetric periodic algebras

Adam Skowyrski

Abstract

Following methods used by A. Dugas for investigating derived equivalent pairs of (weakly) symmetric algebras, we apply them in a specific situation, obtaining new deep results concerning iterated mutations of symmetric periodic algebras. More specifically, for any symmetric algebra $Λ$, and an arbitrary vertex $i$ of its Gabriel quiver, one can define mutation $μ_i(Λ)$ of $Λ$ at vertex $i$ via silting mutation of the stalk complex $\La$. Then $μ_i(Λ)$ is again symmetric, and we can iterate this process. We want to understand the order of $μ_i$, in case the vertex $i$ is $d$-periodic, i.e. the simple module $S_i$ associated to $i$ is periodic of period $d$ (with respect to the syzygy). The main result of this paper shows that then $μ_i$ has order $d-2$, that is $μ_i^{d-2}(Λ)\congΛ$ (modulo socle), under some additional assumption on the (periodic) projective-injective resolution of $S_i$. Besides, we present briefly some consequences concerning arbitrary periodic vertex and give few sugestive examples showing that this property should hold in general, i.e. without restrictions on the periodic projective resolution.

Iterated mutations of symmetric periodic algebras

Abstract

Following methods used by A. Dugas for investigating derived equivalent pairs of (weakly) symmetric algebras, we apply them in a specific situation, obtaining new deep results concerning iterated mutations of symmetric periodic algebras. More specifically, for any symmetric algebra , and an arbitrary vertex of its Gabriel quiver, one can define mutation of at vertex via silting mutation of the stalk complex . Then is again symmetric, and we can iterate this process. We want to understand the order of , in case the vertex is -periodic, i.e. the simple module associated to is periodic of period (with respect to the syzygy). The main result of this paper shows that then has order , that is (modulo socle), under some additional assumption on the (periodic) projective-injective resolution of . Besides, we present briefly some consequences concerning arbitrary periodic vertex and give few sugestive examples showing that this property should hold in general, i.e. without restrictions on the periodic projective resolution.
Paper Structure (7 sections, 9 theorems, 54 equations)

This paper contains 7 sections, 9 theorems, 54 equations.

Table of Contents

  1. Introduction and the main result
  2. Basic notions
  3. Definition of the mutation
  4. Iterating mutation at periodic vertex
  5. Proof of Theorem A
  6. Some examples
  7. Proof of Corollary 4

Key Result

Theorem 2.1

Algebras $\Lambda$ and $\Lambda'$ are derived equivalent if and only if $\Lambda'$ is of the form $\Lambda'\cong\operatorname{End}_{\mathcal{K}^b_\Lambda}(T)$, for a complex $T\in \mathcal{K}^b_\Lambda$ satisfying the following two conditions:

Theorems & Definitions (19)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Remark 2.4
  • Remark 3.1
  • Remark 3.2
  • Theorem 4.1
  • proof
  • Corollary 4.2
  • Definition 4.3
  • ...and 9 more