1-quasi-hereditary algebras: Examples and invariant submodules of projectives

TL;DR

This paper explores the structure of 1-quasi-hereditary algebras, focusing on invariant submodules of projectives and their behavior under Ringel duality, providing new insights and examples in algebra representation theory.

Contribution

It demonstrates that modules generated by a special basis are also modules over endomorphism algebras and characterizes local Delta-good submodules under certain duality conditions.

Findings

Modules generated by basis-elements are also modules over endomorphism algebras.

Characterization of local Delta-good submodules in specific duality cases.

Provides several illustrative examples.

Abstract

In arXiv:1104.4441 it was shown that any 1-quasi-hereditary algebra affords a particular basis which is related to a given partial order on the set of simple modules. We show that the modules generated by these basis-elements are also modules over the endomorphism algebra of some projective indecomposable modules. In case the Ringel-dual of a 1-quasi-hereditary algebra is also 1-quasi-hereditary, we describe all local Delta-good submodules of projective indecomposable modules. We also consider several examples.