Presentations of trivial extensions of finite dimensional algebras and a theorem of Sheila Brenner
Let \(Q\) be a finite connected quiver and \(k\) an algebraically closed field. Consider a finite dimensional algebra \(\Lambda\) of the form \(kQ/I\), where \(I\) is an admissible ideal in the path algebra \(kQ\) of \(Q\). If \(D(\Lambda)\) denotes the dual algebra then the trivial extension of \(\Lambda\) is the algebra \(T(\Lambda)\) equal to \(\Lambda\oplus D(\Lambda)\) as a \(k\)-vector space, equipped with the multiplication given by \((\lambda,f)(\mu,g)=(\lambda\mu,\lambda g+f\mu)\) [see \textit{Y. Iwanaga} and \textit{T. Wakamatsu} in Representation theory II, Lect. Notes Math. 832, 295-301 (1980; Zbl 0445.16024)]. The main results of the paper describe the presentation of \(T(\Lambda)\) and of the repetitive algebra \(\widehat\Lambda\) [see \textit{D. Hughes} and \textit{J. Waschbüsch}, Proc. Lond. Math. Soc., III. Ser. 46, 347-364 (1983; Zbl 0488.16021)] by quivers with relations, under the assumption that every oriented cycle in \(Q\) is zero in \(\Lambda\). A result of \textit{S. Brenner} [Arch. Math. 62, No. 3, 203-206 (1994; Zbl 0796.16016)] is applied in this situation to obtain a characterization of the number of indecomposable direct summands of the middle term of the almost split sequence starting at a simple \(T(\Lambda)\)-module.
scientific article; zbMATH DE number 1750358