Representation dimension of extensions of hereditary algebras

TL;DR

This paper investigates the representation dimension of certain algebra extensions, establishing upper bounds under specific conditions related to the hereditary algebra's type and module properties.

Contribution

It provides new bounds on the representation dimension of triangular matrix algebras derived from hereditary algebras, depending on algebra type and module characteristics.

Findings

Representation dimension ≤ 3 under finite type conditions

Bound holds for tame hereditary algebras with specific modules

No self-extensions in modules implies the bound

Abstract

We show that if H is a hereditary finite dimensional algebra, M is a finitely generated H-module and B is a semisimple subalgebra of the endomorphism algebra of M, then the representation dimension of the corresponding triangular matrix algebra is less or equal to 3 whenever one of the following conditions hold: i) H is of finite representation type; ii) H is tame and M is a direct sum of regular and preprojective modules; iii) M has no self-extensions