Affine highest weight categories and affine quasihereditary algebras

TL;DR

This paper introduces the concepts of affine quasihereditary algebras and affine highest weight categories, extending classical finite-dimensional theories to infinite-dimensional settings, and proves an affine analogue of a key theorem.

Contribution

It axiomatizes affine quasihereditary algebras and affine highest weight categories, establishing foundational theory and an affine version of the Cline-Parshall-Scott Theorem.

Findings

Defined affine quasihereditary algebras and affine highest weight categories.

Proved an affine analogue of the Cline-Parshall-Scott Theorem.

Developed stratified versions of these notions.

Abstract

Koenig and Xi introduced {\em affine cellular algebras}. Kleshchev and Loubert showed that an important class of {\em infinite dimensional} algebras, the KLR algebras $R(\Gamma)$ of finite Lie type $\Gamma$, are (graded) affine cellular; in fact, the corresponding affine cell ideals are idempotent. This additional property is reminiscent of the properties of {\em quasihereditary algebras} of Cline-Parshall-Scott in a {\em finite dimensional} situation. A fundamental result of Cline-Parshall-Scott says that a finite dimensional algebra $A$ is quasihereditary if and only if the category of finite dimensional $A$-modules is a {\em highest weight category}. On the other hand, S. Kato and Brundan-Kleshchev-McNamara proved that the category of {\em finitely generated graded} $R(\Gamma)$-modules has many features reminiscent of those of a highest weight category. The goal of this paper is to axiomatize and study the notions of an {\em affine quasihereditary algebra} and an {\em affine highest weight category}. In particular, we prove an affine analogue of the Cline-Parshall-Scott Theorem. We also develop {\em stratified} versions of these notions.