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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.

MSC:

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.)

Citations:

Zbl 1058.18007
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