Derived Picard Groups of Finite Dimensional Hereditary Algebras 
Jun-ichi Miyachi, Amnon Yekutieli 

24 pages, AMSLaTeX with 6 encapsulated postscript figures: fig0 - fig5.eps
to appear in Compositio Math 


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.
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. 

Math Subject Class: 18E30, 16G20; 16G70, 16G60, 16E20, 14C22