Highest weight categories and stability conditions

TL;DR

This paper introduces new practical criteria for identifying highest weight categories, using numerical and stability condition approaches, which simplify verification and extend to related algebraic structures.

Contribution

It presents two novel criteria—one numerical and one based on stability conditions—for recognizing highest weight categories, improving practical verification and generalizing existing characterisations.

Findings

Numerical criterion based on Grothendieck group simplifies verification.

Stability condition criterion extends to properly stratified categories.

Implication for modules over monomial algebras being highest weight.

Abstract

Highest weight categories are an abstraction of the representation theory of semisimple Lie algebras introduced by Cline, Parshall and Scott in the late 1980s. There are by now many characterisations of when an abelian category is highest weight, but most are hard to verify in practice. We present two new criteria - one numerical in terms of the Grothendieck group, and one in terms of Bridegland stability conditions - which are easier to verify. The stability criterion naturally generalises to a characterisation of properly stratified categories. The numerical criterion implies a criterion of Green and Schroll for when modules over a monomial algebra are highest weight.