Derived categories, tilted algebras, and Drinfelʹd doubles

Given a finite dimensional hereditary algebra Λ over a finite field k, in the derived category Db(Λ) we obtain some formulae on Hall numbers associated to triangles. Using them we prove that, for any tilted algebra Γ of Λ, there is an embedding of the twisted Ringel–Hall algebra of Γ into the Drinfelʹd double of the twisted Ringel–Hall algebra of Λ. Furthermore, if Γ is also hereditary, we show that this embedding can be extended as an isomorphism between the two corresponding Drinfelʹd doubles. These formulae are also used to construct an automorphism of directly associated to Auslander–Reiten translation, and this automorphism follows by B. Sevenhant–M. Van den Berghʹs construction in the case of quivers.