Characterisations of trivial extensions

TL;DR

This paper characterizes trivial extension algebras using quivers with relations and provides a new proof of Wakamatsu's theorem to determine when two algebras have isomorphic trivial extensions.

Contribution

It offers a new characterization of trivial extension algebras and a novel proof of Wakamatsu's theorem based on quiver and relations analysis.

Findings

Explicit description of the ideal of relations for trivial extensions

Characterization of trivial extension algebras via quivers with relations

New proof of Wakamatsu's theorem relating algebra isomorphisms to trivial extensions

Abstract

In this paper we give a characterisation of trivial extension algebras in terms of quivers with relations. This result is based on a explicit description of the ideal of relations of the trivial extension of an algebra, given by the first author in the appendix. We also give a new proof of Wakamatsu's theorem in terms of their quiver and relations, which determines when two given algebras have isomorphic trivial extensions.