1-quasi-hereditary algebras

TL;DR

This paper introduces 1-quasi-hereditary algebras, a subclass of quasi-hereditary algebras inspired by block structures in category O, providing explicit descriptions of their quivers, relations, and tilting modules.

Contribution

It defines 1-quasi-hereditary algebras, explores their properties, and characterizes their structure and tilting modules, especially under Ringel duality.

Findings

Determined quivers and relations for 1-quasi-hereditary algebras.

Characterized the structure of characteristic tilting modules.

Established conditions under which the Ringel dual is also 1-quasi-hereditary.

Abstract

Motivated by the structure of the algebras associated to the blocks of the BGG-category O we define a subclass of quasi-hereditary algebras called 1-quasi-hereditary. Many properties of these algebras only depend on the defining partial order. In particular, we can determine the quiver and the form of the relations. Moreover, if the Ringel dual of a 1-quasi-hereditary algebra is also 1-quasi-hereditary, then the structure of the characteristic tilting module can be computed.