On the correspondence between path algebras and generalized path algebras

TL;DR

This paper extends the theory of generalized path algebras to include relations, providing new methods to identify when a given algebra is isomorphic to such an algebra, thus deepening the understanding of their structure.

Contribution

It generalizes generalized path algebras to incorporate relations and extends existing results to this broader setting, also addressing the inverse problem of algebra isomorphism.

Findings

Extended the concept of generalized path algebras to include relations.

Provided criteria for when an algebra is isomorphic to a generalized path algebra.

Extended previous results on Gabriel quivers to the new setting.

Abstract

The concept of generalized path algebras was introduced in (Coelho and Liu, 2000). It was shown in (Ib\'a\~nez Cobos et al., 2008) how to obtain the Gabriel quiver of a given generalized path algebra. In this article, we generalize the concept of generalized path algebra to allow them to have relations, and we extend the result in (Ib\'a\~nez Cobos et al., 2008) to this new setting. Moreover, we use the extended result mentioned above to address the inverse problem: that is, the problem of determining when a given algebra is isomorphic to a generalized path algebra in a non-trivial way.