Tilted algebras and crossed products (Q2816779)

Let \(A\) be a connected Artinian algebra, \(G\) a finite group, \(\alpha\) a 2-cocyle, \(\sigma:G\to\mathrm{Aut}(A)\) a homomorphism, \(A{}_{\alpha}\#_{\sigma}G\) the crossed product of \(A\) and \(G\). There is a recent invigoration in the study of representation theory over crossed product algebras, see for instance, [\textit{L. F. Barannyk} and \textit{D. Klein}, Commun. Algebra 42, No. 9, 4131--4147 (2014; Zbl 1317.16023); J. Algebra 403, 300--312 (2014; Zbl 1310.16020)]. The authors in the present paper consider the classic result of \textit{I. Reiten} and \textit{C. Riedtmann} [J. Algebra 92, 224--282 (1985; Zbl 0549.16017)]: \(A\) is a representation finite tilted algebra if and only if so is \(A{}_{\alpha}\#_{\sigma}G\), and they extend it as follows. \(A\) is tilted if and only if so is \(A{}_{\alpha}\#_{\sigma}G\). The proof applies the characterization of tiltedness by means of the standard component \(C_A\) of the Auslander-Reiten quiver of \(A\), and the description of the component \(C_A\). An example for the application of the main result is provided in the representation-infinite case.