## 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.)

Zbl 1058.18007
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