Quasi-hereditary algebras via generator-cogenerators of local self-injective algebras and transfer of Ringel duality

TL;DR

This paper explores the structure of 1-quasi-hereditary algebras with dominant dimension at least two, relating them to certain bimodule classes and examining how Ringel duality affects their properties.

Contribution

It characterizes the class B of algebras connected to 1-quasi-hereditary algebras via bimodules and analyzes the transfer of Ringel duality within this framework.

Findings

Dominant dimension of Ringel duals is at least two.

Class B of algebras is specified and related to 1-quasi-hereditary algebras.

Ringel duality does not preserve the class A of 1-quasi-hereditary algebras.

Abstract

The dominant dimension of algebras in the class A of 1-quasi-hereditary algebras is at least two. By the Morita-Tachikawa Theorem this implies that A is related to a certain class B of algebras via bimodules satisfying the double centralizer condition. In this paper we specify the class B and the modules over algebras in B connected with A. The class A is not closed under taking the Ringel-dual. However the dominant dimension of the Ringel-dual R(q) of a 1-quasi-hereditary algebra q is at least two. This fact induces a corresponding concept of modules over algebras in B which yield the algebras q and R(q) for q in A.