Derived Picard Groups of Finite Dimensional Hereditary Algebras

Details Derived Picard Groups of Finite Dimensional Hereditary Algebras Derived Picard Groups of Finite Dimensional Hereditary Algebras

1999-04-02

arxiv.org/abs/math/9904006v3

Mathematics

Rings and Algebras

24 pages, AMSLaTeX with 6 encapsulated postscript figures, some errors corrected, introductory material edited, to appear in C

Scientific paper

Let A be an algebra over a field k, and denote by D^b(Mod A) the bounded derived category of left A-modules. The derived Picard group DPic_k(A) is the group of triangle auto-equivalences of D^b(Mod A) induced by tilting complexes. We study the group DPic_k(A) when A = k \Delta is the path algebra of a finite quiver \Delta. We obtain general results on the structure of DPic_k(A), as well as explicit calculations for many cases, including all finite and tame representation types. Our method is to construct a representation of DPic_k(A) on a certain infinite quiver. This representation is faithful when\Delta is a tree, and then DPic_k(A) is discrete. Otherwise a connected linear algebraic group can occur as a factor of DPic_k(A). When A is hereditary, DPic_k(A) coincides with the full group of k-linear triangle auto-equivalences of D^b(Mod} A). Hence we can calculate the group of such auto-equivalences for any triangulated category D equivalent to D^b(Mod A). These include the derived categories of certain noncommutative spaces introduced by Kontsevich-Rosenberg.

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