Folded tilting complexes for Brauer tree algebras.
Let \(A\) and \(B\) be two \(k\)-algebras, projective as \(k\)-modules for an integral domain \(k\). Then, the derived categories of \(A\)-modules and the derived category of \(B\)-modules are equivalent if and only if there is a so-called tilting complex \(T\) of \(A\)-modules so that \(B\) is the endomorphism ring of \(T\). The complex \(T\) is the image of the rank one free \(B\)-module under the equivalence. Among the very first examples studied for this theory were Brauer tree algebras. These are algebras associated to a finite graph with additional combinatorial data, a Brauer tree. Using an idea of \textit{S. König} and the reviewer [Commun. Algebra 24, No. 6, 1897--1913 (1996; Zbl 0851.16012)], \textit{M. Schaps} and \textit{E. Zakay-Illouz} constructed a combinatorially defined complex \(T\) over the Brauer tree algebra \(A\) associated to a star with endomorphism ring being any of the algebras amongst those which are derived equivalent to the given one [J. Algebra 246, No. 2, 647--672 (2001; Zbl 0994.16011)], so that \(T=F(A)\) for an equivalence \(F:D^b(B)\rightarrow D^b(A)\). In the paper under review a complex \(S\) is constructed corresponding to \(G(A)\) for a quasi-inverse \(G\) of \(F\).
scientific article; zbMATH DE number 1888321