Derived Picard groups of finite-dimensional hereditary algebras Academic Article uri icon

abstract

  • 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 …

publication date

  • January 1, 2001