Stability Conditions and Harder-Narasimhan Filtrations for Triangulated Categories

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 $D^b(\mathrm{rep} A_2)$.