Given a finite dimensional hereditary algebra Lambda over a finite field k, in the derived category D-b(Lambda) we obtain some formulae on Hall numbers associated to triangles. Using them we prove that. for any tilted algebra Gamma of Lambda, there is an embedding of the twisted Ringel-Hall algebra of Gamma into the Drinfel'd double D(Lambda) of the twisted Ringel-Hall algebra of Lambda. Furthermore, if Gamma is also hereditary, we show that this embedding can be extended as an isomorphism between the two corresponding Drinfel'd doubles. These formulae are also used to construct an automorphism of D(Lambda) directly associated to Auslander-Reiten translation, and this amomorphism follows by B. Sevenhant-M. Van den Bergh's construction in the case of quivers. (C) 2003 Elsevier Inc. All rights reserved. [References: 12]
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