In the bocs seat: Quasi-hereditary algebras and representation type

TL;DR

This paper explores the relationship between bocses and quasi-hereditary algebras, emphasizing their impact on understanding the representation type of module categories, including new classification results and applications to tame Schur algebras.

Contribution

It provides a survey of recent connections between bocses and quasi-hereditary algebras, introduces new proofs, and applies these to classify representation types of certain module categories.

Findings

Modules filtered by Weyl modules for tame Schur algebras are of finite representation type.

A new proof classifies quasi-hereditary algebras with two simple modules.

The relationship between bocses and quasi-hereditary algebras impacts representation theory.

Abstract

This paper surveys bocses, quasi-hereditary algebras and their relationship which was established in a recent result by Koenig, Ovsienko, and the author. Particular emphasis is placed on applications of this result to the representation type of the category filtered by standard modules for a quasi-hereditary algebra. In this direction, joint work with Thiel is presented showing that the subcategory of modules filtered by Weyl modules for tame Schur algebras is of finite representation type. The paper also includes a new proof for the classification of quasi-hereditary algebras with two simple modules, a result originally obtained by Membrillo-Hern\'andez.