Peng, Liangang; Xiao, Jie Invariability of repetitive algebras of tilted algebras under stable equivalence. (English) Zbl 0824.16014 J. Algebra 170, No. 1, 54-68 (1994). Let \(A\) be a finite dimensional \(K\)-algebra over an algebraically closed field \(K\). The trivial extension \(A \ltimes D(A)\) of \(A\) by the injective cogenerator \(D(A) = \operatorname{Hom}_K (A,K)\) admits a canonical Galois covering \(\widehat {A} \to A \ltimes D(A)\) with infinite cyclic group, where \(\widehat {A}\) is the repetitive (locally bounded, Frobenius) algebra of \(A\). If \(A\) is of finite global dimension, then by a result due to D. Happel [Comment. Math. Helv. 62, 339–389 (1987; Zbl 0626.16008)] the stable module category \(\underline {\text{mod}} \widehat {A}\) is equivalent, as a triangulated category, to the derived category \(D^b (A)\) of bounded complexes of finite dimensional \(A\)-modules. In the paper it is shown that if \(\mathcal A\) is a Frobenius locally bounded algebra and there is an equivalence \(\underline \mod \mathcal A} \simeq D^b (K \Delta)\), for some finite quiver \(\Delta\) without oriented cycles, then \({\mathcal A} \simeq \widehat {A}\) for some tilted algebra \(A\) of type \(\Delta\). For \(\Delta\) non-Dynkin, the above algebra \(A\) can be chosen to be representation-infinite. As a consequence one obtains that if \(B\) is an iterated tilted algebra of type \(\Delta\) then there exists a tilted algebra \(A\) of type \(\Delta\) such that \(\widetilde {B} \ltimes \widetilde {A}\) and \(B \ltimes D(B) \simeq A \ltimes D(A)\). Moreover, it follows that a finite dimensional \(K\)-algebra \(B\) is iterated tilted of type \(\Delta\) if and only if \(B\) can be obtained from a tilted algebra \(A\) of type \(\Delta\) by a sequence of reflections in the sense of D. Hughes and J. Waschbüsch [Proc. Lond. Math. Soc. (3) 46, 347–364 (1983; Zbl 0488.16021)]. In the Dynkin case, the above results were proved by Hughes-Waschbüsch and Assem-Happel-Roldan, and in the Euclidean case by Assem-Nehring-Skowroński, Happel, Skowroński. Reviewer: Andrzej Skowroński (Toruń) Cited in 1 ReviewCited in 3 Documents MSC: 16G60 Representation type (finite, tame, wild, etc.) of associative algebras 16G20 Representations of quivers and partially ordered sets 18E30 Derived categories, triangulated categories (MSC2010) 16D90 Module categories in associative algebras 16L60 Quasi-Frobenius rings Keywords:repetitive algebras; tilted algebras; representation-infinite algebras; finite dimensional algebras; trivial extensions; injective cogenerators; Galois coverings; finite global dimension; stable module category; triangulated category; derived category; finite dimensional \(A\)-modules; Frobenius locally bounded algebras; equivalence of categories Citations:Zbl 0503.16026; Zbl 0626.16008; Zbl 0488.16021 PDFBibTeX XMLCite \textit{L. Peng} and \textit{J. Xiao}, J. Algebra 170, No. 1, 54--68 (1994; Zbl 0824.16014) Full Text: DOI