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Triangulated structures induced by mutations

Ryota Iitsuka

TL;DR

This work unifies several mutation paradigms in representation theory by introducing premutation and mutation triples within extriangulated categories. It proves that premutation triples naturally yield pretriangulated structures on the quotient $\mathcal{Z}/[\mathcal{I}]$, and identifies an MT4-type condition under which these become triangulated categories, extending known results from Frobenius, concentric twin cotorsion pairs, and simple-minded mutation theories. The framework provides right and left mutation doubles to generate robust triangulated structures and clarifies how restriction to extension-closed subcategories preserves the mutational behavior, connecting silting, cluster-tilting, and simple-minded mutations in a single setting. By organizing reduction theory via reducible triples, the paper also shows how to restrict mutations and to derive new triangulated categories from existing ones, with broad applicability to tilting/silting theory and simple-minded systems. Overall, the results offer a cohesive, extensible toolkit for constructing and comparing triangulated structures arising from various mutation processes in triangulated and extriangulated contexts.

Abstract

In representation theory of algebras, there exist two types of mutation pairs: rigid type (cluster-tilting mutations by Iyama-Yoshino) and simple-minded type (mutations of simple-minded systems by Simões-Pauksztello). It is known that such mutation pairs induce triangulated categories, however, these facts have been proved in different ways. In this paper, we introduce the concept of ''mutation triples'', which is a simultaneous generalization of two different types of mutation pairs as well as concentric twin cotorsion pairs. We present two main theorems concerning mutation triples. The first theorem is that mutation triples induce pretriangulated categories. The second one is that pretriangulated categories induced by mutation triples become triangulated categories if they satisfy an additional condition (MT4).

Triangulated structures induced by mutations

TL;DR

This work unifies several mutation paradigms in representation theory by introducing premutation and mutation triples within extriangulated categories. It proves that premutation triples naturally yield pretriangulated structures on the quotient , and identifies an MT4-type condition under which these become triangulated categories, extending known results from Frobenius, concentric twin cotorsion pairs, and simple-minded mutation theories. The framework provides right and left mutation doubles to generate robust triangulated structures and clarifies how restriction to extension-closed subcategories preserves the mutational behavior, connecting silting, cluster-tilting, and simple-minded mutations in a single setting. By organizing reduction theory via reducible triples, the paper also shows how to restrict mutations and to derive new triangulated categories from existing ones, with broad applicability to tilting/silting theory and simple-minded systems. Overall, the results offer a cohesive, extensible toolkit for constructing and comparing triangulated structures arising from various mutation processes in triangulated and extriangulated contexts.

Abstract

In representation theory of algebras, there exist two types of mutation pairs: rigid type (cluster-tilting mutations by Iyama-Yoshino) and simple-minded type (mutations of simple-minded systems by Simões-Pauksztello). It is known that such mutation pairs induce triangulated categories, however, these facts have been proved in different ways. In this paper, we introduce the concept of ''mutation triples'', which is a simultaneous generalization of two different types of mutation pairs as well as concentric twin cotorsion pairs. We present two main theorems concerning mutation triples. The first theorem is that mutation triples induce pretriangulated categories. The second one is that pretriangulated categories induced by mutation triples become triangulated categories if they satisfy an additional condition (MT4).
Paper Structure (27 sections, 86 theorems, 128 equations)

This paper contains 27 sections, 86 theorems, 128 equations.

Table of Contents

  1. Introduction
  2. Structures associated with additive category
  3. Extriangulated categories
  4. Pretriangulated categories
  5. Pretriangulated structures induced by premutation triples
  6. The condition (MT1) and (MT2)
  7. Definition of premutation triples
  8. Right triangles induced by a right mutation double
  9. Right triangulated structures induced by right mutation doubles
  10. Pretriangulated structures induced by premutation triples
  11. Triangulated structures induced by mutation triples
  12. Mutation triples and main theorem
  13. Comparisons with known results in concentric twin cotorsion pairs
  14. Reducible triples
  15. (Pre)triangulated structures induced by concentric twin cotorsion pairs
  16. ...and 12 more sections

Key Result

Theorem 1.2

(Theorem main_thm1) Let $(\mathcal{S}, \mathcal{Z}, \mathcal{V})$ be a premutation triple. Then $\mathcal{Z}/[\mathcal{I}]$ has a pretriangulated structure.

Theorems & Definitions (272)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 1.5
  • Definition 1.6
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • ...and 262 more