**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