Derived Picard Groups of Finite-Dimensional Hereditary Algebras

Abstract

Let A be a finite-dimensional algebra over a field k. The derived Picard group DPic_k(A) is the group of triangle auto-equivalences of D^b(mod A) induced by two-sided tilting complexes. We study the group DPic_k(A) when A is hereditary and k is algebraically closed. We obtain general results on the structure of DPic_k, 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 Γ^irr. This representation is faithful when the quiver Δ of A 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 piecewise hereditary algebras, and of certain noncommutative spaces introduced by Kontsevich and Rosenberg.