  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.
