Silting mutation in triangulated categories

TL;DR

This paper introduces silting mutation as a generalization of tilting mutation in triangulated categories, overcoming previous limitations and establishing a foundational theory with applications to algebraic structures.

Contribution

It develops a basic theory of silting mutation, including a partial order on silting objects and its relation to mutation, extending existing mutation theories.

Findings

Silting mutation generalizes tilting mutation.

Iterated silting mutation acts transitively on silting objects for certain algebras.

A bijection between silting subcategories and specific t-structures is established.

Abstract

In representation theory of algebras the notion of `mutation' often plays important roles, and two cases are well known, i.e. `cluster tilting mutation' and `exceptional mutation'. In this paper we focus on `tilting mutation', which has a disadvantage that it is often impossible, i.e. some of summands of a tilting object can not be replaced to get a new tilting object. The aim of this paper is to take away this disadvantage by introducing `silting mutation' for silting objects as a generalization of `tilting mutation'. We shall develope a basic theory of silting mutation. In particular, we introduce a partial order on the set of silting objects and establish the relationship with `silting mutation' by generalizing the theory of Riedtmann-Schofield and Happel-Unger. We show that iterated silting mutation act transitively on the set of silting objects for local, hereditary or canonical algebras. Finally we give a bijection between silting subcategories and certain t-structures.