Triangulated structures induced by mutations

TL;DR

This paper introduces mutation triples, a unifying concept in algebra representation theory, showing they induce pretriangulated categories that become triangulated under certain conditions, linking different mutation types.

Contribution

It generalizes two mutation pair types into mutation triples and proves their induced categories are pretriangulated, becoming triangulated with an extra condition.

Findings

Mutation triples induce pretriangulated categories.

Additional conditions make these categories triangulated.

Unifies different mutation pair frameworks.

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 Sim\~oes-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).