Foldings and two-sided tilting complexes for Brauer tree algebras

Abstract

In this note, for a Brauer tree algebra $A$ and a star-shaped Brauer tree algebra $B$ which is derived equivalent to $A$, we give operations on the two-sided tilting complex $D_T$ of $A\otimes B^{op}$-modules constructed in [3] which is isomorphic to the Rickard tree-to-star complex $T$ constructed in [5] in $D^b(A)$, and we show that the operations on $D_T$ correspond to operations called $foldings$ on the Rickard tree-to-star complex $T$ given in [7].