Derived Picard Groups of Finite Dimensional Hereditary Algebras

الأصل · EN

Let A be an algebra over a field k, and denote by Dᵇ(Mod A) the bounded derived category of left A-modules. The derived Picard group DPicₖ(A) is the group of triangle auto-equivalences of Dᵇ(Mod A) induced by tilting complexes. We study the group DPicₖ(A) when A = k Δis the path algebra of a finite quiver Δ. We obtain general results on the structure of DPicₖ(A), as well as explicit calculations for many cases, including all finite and tame representation types. Our method is to construct a representation of DPicₖ(A) on a certain infinite quiver. This representation is faithful whenΔis a tree, and then DPicₖ(A) is discrete. Otherwise a connected linear algebraic group can occur as a factor of DPicₖ(A). When A is hereditary, DPicₖ(A) coincides with the full group of k-linear triangle auto-equivalences of Dᵇ(Mod A). Hence we can calculate the group of such auto-equivalences for any triangulated category D equivalent to Dᵇ(Mod A). These include the derived categories of certain noncommutative spaces introduced by Kontsevich-Rosenberg.