Isomorphic trivial extensions of finite dimensional algebras. (Q2576912)

Let \(T(\Lambda)\) denote the trivial extension of an Artin algebra \(\Lambda\) by its minimal injective cogenerator. The question when \(T(\Lambda)\) is isomorphic to \(T(\Lambda')\) was studied by \textit{T. Wakamatsu} [Commun. Algebra 12, 33-41 (1984; Zbl 0537.16008)]. The authors of the present paper consider this problem for algebras over an algebraically closed field. The aim is to classify in terms of quivers and relations all algebras \(\Lambda'\) such that \(T(\Lambda')\cong T(\Lambda)\) for a given \(\Lambda\). Such a classification is obtained under the assumption that every oriented cycle in the ordinary quiver of \(\Lambda\) is zero in \(\Lambda\).