The classification of tilting modules over Harada algebras

TL;DR

This paper classifies certain modules and tilting modules over Harada algebras, linking them to tilting modules over upper triangular matrix algebras, thereby providing a comprehensive understanding of their structure.

Contribution

It introduces a classification of modules with projective dimension at most one and establishes a bijection between tilting modules over Harada algebras and those over direct products of upper triangular matrix algebras.

Findings

Classified modules with projective dimension ≤ 1 over Harada algebras.

Established a bijection between tilting modules over Harada and upper triangular matrix algebras.

Provided a combinatorial description of tilting modules over upper triangular matrix algebras.

Abstract

In the 1980s, Harada introduced a new class of algebras now called Harada algebras. Harada algebras provides us with a rich source of Auslander's 1-Gorenstein algebras. In this paper, we have two main results about Harada algebras. The first is the classification of modules over Harada algebras whose projective dimension is at most one. The second is the classification of tilting modules over Harada algebras, which is shown by giving a bijection between tilting modules over Harada algebras and tilting modules over direct products of upper triangular matrix algebras over $K$. A combinatorial description of tilting modules over upper triangular matrix algebras over $K$ is known. These facts allow us to classify tilting modules over a given Harada algebra.