Highest weight categories and stability conditions

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.

💡 Research Summary

The paper “Highest weight categories and stability conditions” by A. Cipriani and J. Woolf introduces two new, practically verifiable criteria for recognizing when an abelian category is a highest‑weight category. The authors work throughout with a Deligne‑finite k‑linear abelian category A (k algebraically closed) equipped with a total ordering of its finitely many simple objects S₁,…,Sₙ. Standard objects Δᵢ and costandard objects ∇ᵢ are defined as the maximal quotients of the projective covers Pᵢ and maximal subobjects of the injective hulls Iᵢ lying in the Serre subcategory generated by the first i simples.

The first main result (Theorem 1) is a purely numerical criterion expressed in the Grothendieck group K(A). It states that A is highest‑weight if and only if the class of each standard object satisfies