Sonia Trepode (Mar del Plata), Characterisations of trivial extension algebras.

Abstract:

In this talk we consider the trivial extension of basic connected finite dimensional algebras where, by trivial extension, we always mean the trivial extension by the minimal injective co-generator.

In general, deciding whether a given algebra is a trivial extension or not is not an easy task. In this talk we answer this question by giving, in terms of its quiver and relations, a complete characterisation of the trivial extension of a finite dimensional algebra over a field. Our theorem provides an algorithm to decide when a finite dimensional algebra is a trivial extension or not.

Fernández and Platzeck introduced some cutting sets, that we call admissible cuts, in order to study isomorphic trivial extensions in particular cases. Using their techniques and the tools of split by nilpotent extensions, we are able to prove this result in general.

We characterise the algebras having isomorphic trivial extensions in terms of admissible cuts. Fernández and Platzeck also gave a diagrammatic interpretation of Wakamatsu's theorem for isomorphic trivial extensions under some assumptions. Following these ideas, and using the tools of split by nilpotent extensions, we are able to give an independent proof of Wakamatsu's result for finite dimensional algebras.

Joint work with Fernández, Elsa; Schroll, Sibylle; Treffinger, Hipólito and Valdivieso, Yadira.