Based quasi-hereditary algebras

TL;DR

This paper enhances the basis properties of split quasi-hereditary algebras over complete local Noetherian rings, facilitating their 'schurification' into new quasi-hereditary and cellular algebras with super-structures.

Contribution

It develops a stronger basis approach for split quasi-hereditary algebras over complete local rings, enabling their schurification while preserving key properties.

Findings

Achieved stronger basis properties for quasi-hereditary algebras

Established conditions for schurification to preserve quasi-heredity

Proved Morita equivalence results for basic quasi-hereditary algebras

Abstract

A notion of a split quasi-hereditary algebra has been defined by Cline, Parshall and Scott. Du and Rui describe a based approach to split quasi-hereditary algebras. We develop this approach further to show that over a complete local Noetherian ring, one can achieve even stronger basis properties. This is important for `schurifying' quasi-hereditary algebras as developed in our subsequent work. The schurification procedure associates to an algebra $A$ a new algebra, which is the classical Schur algebra if $A$ is a field. Schurification produces interesting new quasi-hereditary and cellular algebras. It is important to work over an integral domain of characteristic zero, taking into account a super-structure on the input algebra $A$. So we pay attention to super-structures on quasi-hereditary algebras and investigate a subtle conforming property of heredity data which is crucial to guarantee that the schurification of $A$ is quasi-hereditary if so is $A$. We establish a Morita equivalence result which allows us to pass to basic quasi-hereditary algebras preserving conformity.