Stability Conditions and Harder-Narasimhan Filtrations for Triangulated Categories

TL;DR

This paper explores stability conditions and filtrations in triangulated categories, extending known results from abelian categories, and characterizes stability conditions on a specific derived category.

Contribution

It generalizes Harder-Narasimhan filtrations and wall-crossing formulas from abelian to triangulated categories, providing new insights and characterizations.

Findings

Extended Reineke inversions to triangulated categories

Derived Hall algebra relationships established

Characterized all stability conditions on D^b(rep A_2)

Abstract

In this paper, we investigate the relationships between Harder-Narasimhan filtrations and derived Hall algebras. We extend several results from abelian categories to triangulated categories, including Reineke inversions, wall-crossing formulas, and Joyce's elements $\epsilon_\gamma$. The results in triangulated categories can be summarized via a diagram of the same form of that in abelian categories. As an application, we characterize all possibilities for stability conditions on $D^b(\mathrm{rep} A_2)$.