Maximal green sequences for preprojective algebras

TL;DR

This paper extends the concept of maximal green sequences from quivers to hearts of bounded t-structures in triangulated categories, characterizing when such sequences exist for preprojective algebras and finite-dimensional algebras.

Contribution

It introduces a categorical framework for maximal green sequences applicable to hearts of t-structures and characterizes their existence for Dynkin and non-Dynkin quivers.

Findings

Maximal green sequences exist if and only if the quiver is of Dynkin type.

Maximal chains in the Hasse quiver correspond to maximal green sequences.

The framework applies to module categories with finitely many bricks.

Abstract

Maximal green sequences were introduced as combinatorical counterpart for Donaldson-Thomas invariants for 2-acyclic quivers with potential by B. Keller. We take the categorical notion and introduce maximal green sequences for hearts of bounded t-structures of triangulated categories that can be tilted indefinitely. We study the case where the heart is the category of modules over the preprojective algebra of a quiver without loops. The combinatorical counterpart of maximal green sequences for Dynkin quivers are maximal chains in the Hasse quiver of basic support \tau -tilting modules. We show that a quiver has a maximal green sequence if and only if it is of Dynkin type. More generally, we study module categories for finite- dimensional algebras with finitely many bricks.