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Highest weight categories and stability conditions

Alessio Cipriani, Jon Woolf

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.

Highest weight categories and stability conditions

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.

Paper Structure

This paper contains 7 sections, 7 theorems, 27 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Highest-weight categories
  3. Stability functions
  4. Two new characterisations of highest-weight categories
  5. Properly stratified categories
  6. Green--Schroll's sufficient criterion
  7. Examples

Key Result

Lemma 2.2

The standard object ${\Delta_{i}}$ is a projective cover, and the costandard object ${\nabla_{\! i}}$ an injective hull, of $S_i$ in $\mathcal{A}_i$.

Figures (1)

  • Figure 1: Auslander-Reiten quivers of the algebras from Example \ref{['ex:1']} (top) and Example \ref{['ex:3']} (bottom).

Theorems & Definitions (23)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Theorem 2.5
  • proof
  • Remark 2.6
  • Theorem 2.7: PS, MR0987824, Cline1988
  • Lemma 3.1
  • proof
  • ...and 13 more