Presentations of trivial extensions of finite dimensional algebras and a theorem of Sheila Brenner

Abstract

Let Λ be a finite dimensional algebra over an algebraically closed field such that any oriented cycle in the ordinary quiver of Λ is zero in Λ. We describe the ordinary quiver and relations for T(Λ) = Λ ⋉ D(Λ), the trivial extension of Λ by its minimal injective cogenerator D(Λ), and also for the repetitive algebra Λ of Λ. Associated with this description we give an application of a theorem of Sheila Brenner. © 2002 Elsevier Science (USA).