On Moduli Spaces for Abelian Categories

TL;DR

This paper develops a framework for constructing moduli spaces of (semi)stable objects in certain abelian categories, providing a geometric perspective on their classification, with applications to categories in representation theory.

Contribution

It introduces the concept of moduli spaces for (semi)stable objects in abelian categories under mild conditions, linking algebraic and geometric structures.

Findings

Moduli spaces can be constructed as projective algebraic varieties.

Application to highest weight categories in representation theory.

Framework generalizes existing moduli space concepts.

Abstract

We show that if A is an abelian category satisfying certain mild conditions, then one can introduce the concept of a moduli space of (semi)stable objects which has the structure of a projective algebraic variety. This idea is applied to several important abelian categories in representation theory, like highest weight categories.