On strongly quasi-hereditary algebras

TL;DR

This paper studies a special class of finite dimensional algebras called left strongly quasi-hereditary algebras, constructing heredity chains and showing certain quotient algebras retain this property.

Contribution

It constructs a heredity chain for left strongly quasi-hereditary algebras and proves quotient algebras by specific ideals are also left strongly quasi-hereditary.

Findings

Constructed a special heredity chain for these algebras.

Proved quotient algebras by certain ideals are also left strongly quasi-hereditary.

Enhanced understanding of the structure and stability of left strongly quasi-hereditary algebras.

Abstract

Let $A$ be a finite dimensional algebra over an algebraically closed field $\mathbf{k}$. If $A$ is quasi-hereditary and the projective dimensions of all standard modules are at most one, then $A$ is called left strongly quasi-hereditary. In this paper, we construct a special heredity chain for left strongly quasi-hereditary algebras. Moreover, we show the quotient algebra by an ideal which appears in a special heredity chain of left strongly quasi-hereditary algebra is also left strongly quasi-hereditary algebra.