Mutation classes of \(\widetilde A_n\)-quivers and derived equivalence classification of cluster tilted algebras of type \(\widetilde A_n\).
Let \(Q\) be a quiver which is mutation equivalent to a quiver of type \(\widetilde A_n\). When referring to type \(\widetilde A_n\) quivers in this review, we always assume that the orientation is not cyclic. This paper identifies four combinatorial parameters, which can be read off directly from a drawing of \(Q\), that determine its mutation class. The basic idea is that each oriented 3-cycle, and each arrow that is not contained in an oriented 3-cycle, are assigned a certain parity according to some rules (solid or dashed). The combinatorial parameters \(r_1\) and \(r_2\) count the number of solid arrows and 3-cycles, respectively, while \(s_1\) and \(s_2\) count the dashed ones. From these, one can classify derived equivalence classes of cluster tilted algebras of type \(\widetilde A_n\).NEWLINENEWLINE The proof builds on the classification of quivers mutation equivalent to a quiver of type \(A\) by \textit{A. B. Buan} and \textit{D. F. Vatne} [J. Algebra 319, No. 7, 2723-2738 (2008; Zbl 1155.16010)]. The author defines a class of quivers \(\mathcal Q_n\) that contains all quivers of type \(\widetilde A_n\), and prove that it is closed under mutation. Therefore \(\mathcal Q_n\) contains all quivers mutation equivalent to one of type \(\widetilde A_n\). Then the author gives an algorithm to take any quiver in this class and reduce it to a normal form, without changing the parameters \(r_1,r_2,s_1,s_2\) at any step. Finally, it is shown that a quiver in normal form is mutation equivalent to a quiver of type \(\widetilde A_n\) with \(r_1+2r_2\) arrows in one direction and \(s_1+2s_2\) arrows in the other direction. Thus \(\mathcal Q_n\) is exactly the set of quivers mutation equivalent to some quiver of type \(\widetilde A_n\).NEWLINENEWLINE Turning the focus to cluster tilted algebras, the main result is proven: namely, that two cluster tilted algebras of type \(\widetilde A_n\) are derived equivalent if and only if they have the same parameters \(r_1,r_2,s_1,s_2\), up to interchanging the roles of \(r\) and \(s\). The strategy of the proof is to use Rickard's result that derived equivalences are given by tilting complexes, and define a tilting complex for each step in the algorithm above. This shows that two algebras whose quivers have the same combinatorial invariant are derived equivalent. The converse employs an invariant of derived equivalent gentle algebras from \textit{D. Avella-Alaminos} and \textit{C. Geiss} [J. Pure Appl. Algebra 212, No. 1, 228-243 (2008; Zbl 1143.16016)].
scientific article; zbMATH DE number 6054514