Linearity of stability conditions

TL;DR

This paper explores the equivalence between linear and nonlinear stability conditions for modules over hereditary algebras, linking concepts like Harder-Narasimhan stratifications and maximal green sequences.

Contribution

It establishes the equivalence of various stability concepts for hereditary algebras and sets the stage for classifying maximal green sequences of affine type A quivers.

Findings

Finite Harder-Narasimhan stratifications are equivalent to finite nonlinear stability conditions.

Maximal forward hom-orthogonal sequences correspond to maximal green sequences.

The paper initiates classification of maximal green sequences for affine type A quivers.

Abstract

For modules over an artin algebra a linear stability condition is given by a "central charge" and a nonlinear stability condition is given by the wall-crossing sequence of a "green path". Finite Harder-Narasimhan stratifications of the module category, maximal forward hom-orthogonal sequences and maximal green sequences, defined using Fomin-Zelevinsky quiver mutation are shown to be equivalent to finite nonlinear stability conditions when the algebra is hereditary. This is the first of a series of three papers whose purpose is to determine all maximal green sequences of maximal length for quivers of affine type A and determine which are linear. See [1].