Publication status: Compositio Mathematica 129 (2001), 341-368.
Abstract:
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.
In [Ye2] we proved that DPic_k(A) parameterizes the isomorphism classes
of
dualizing complexes over A. Also when A is either commutative or local,
DPic_k(A) Pic_k(A) \times Z, where Pic_k(A) is the noncommutative Picard
group
(the group of Morita equivalences).
In this paper 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 the Dynkin and affine
quivers, and for some wild quivers with multiple arrows.
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).
In addition we prove that 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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updated 11.6.02