Derived Picard groups of finite-dimensional hereditary algebras (Q2769564)

Let \(A\) be an algebra over a field \(k\) and let \(D^b(A)\) be the derived category of bounded complexes of \(A\)-modules. Then, the group \(D\text{Pic}_k(A)\) is defined as the group of isomorphism classes of self-equivalences of \(D^b(A)\) given by the derived tensor product of a complex of \(A\otimes_kA^{\text{op}}\)-bimodules. Observe that this group is denoted by \(\text{TrPic}_k(A)\) by Rouquier and the reviewer [CMS Conf. Proc. 18, 721--749 (1996; Zbl 0855.16015)]. In the paper under review the case of a finite dimensional hereditary \(k\)-algebra \(A\) and algebraically closed field \(k\) is studied.NEWLINENEWLINENEWLINELet \(\Gamma^{\text{irr}}\) be the union of the non regular components of the Auslander-Reiten quiver of \(D^b(A)\). The group of automorphisms modulo inner automorphisms of \(A\) is an algebraic group and the identity component \(\text{Out}^\circ_k(A)\) of this group is a normal subgroup of \(D\text{PiC}_k(A)\). The main result of the paper under review is the following. The group \(D\text{Pic}_k(A)\) is a semidirect product of \(\text{Out}^\circ_k(A)\) acted upon by the subgroup of those automorphisms \(\Gamma^{\text{irr}}\) which commute with the action of Auslander-Reiten translation and degree shift on \(\Gamma^{\text{irr}}\). Finally, the authors study the case of finite representation type and the case of tame representation type in complete detail.