Self-equivalences of the derived category of Brauer tree algebras with exceptional vertex. (English) Zbl 1034.16020

Summary: Let \(k\) be a field and \(A\) be a Brauer tree algebra associated with a Brauer tree with possibly non trivial exceptional vertex. In an earlier joint paper with R. Rouquier [Proc. Lond. Math. Soc., III. Ser. 87, No. 1, 197-225 (2003; Zbl 1058.18007)] we studied and defined the group \(\text{Tr\,Pic}_k(\Lambda)\) of standard self-equivalence of the derived category of a \(k\)-algebra \(\Lambda\). In the present note we determine a non-trivial homomorphism of a group slightly bigger than the pure braid group on \(n+1\) strings to \(\text{Tr\,Pic}_k(\Lambda)\). This is a generalization of the main result in the paper mentioned above, and the proof uses the result in the same paper.


16G30 Representations of orders, lattices, algebras over commutative rings
18E30 Derived categories, triangulated categories (MSC2010)
16D90 Module categories in associative algebras
20F36 Braid groups; Artin groups
16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)


Zbl 1058.18007
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