Grobner-Shirshov Basis of Derived Hall Algebra of Type An

We know that in Ringel-Hall algebra of Dynkin type,the set of all skew commutator relations between the iso-classes of indecomposable modules forms a minimal Grobner-Shirshov basis,and the corresponding irreducible elements forms a PBW type basis of the Ringel-Hall algebra.We aim to generalize this result to the derived Hall algebra DH(An)of type An.First,we compute all skew commutator relations between the iso-classes of indecomposable objects in the bounded derived category D^b(An)using the Auslander-Reiten quiver of D^b(An),and then we prove that all possible compositions between these skew commutator relations are trivial.As an application,we give a PBW type basis of DH(An).

Hall Algebras Associated to Complexes of Fixed Size

Let A be a finitary hereditary abelian category with enough projectives.We study the Hall algebra of complexes of fixed size over projectives.Explicitly,we first give a relation between Hall algebras of complexes of fixed size and cyclic complexes.Second,we characterize the Hall algebra of complexes of fixed size by generators and relations,and relate it to the derived Hall algebra of A.Finally,we give the integration map on the Hall algebra of 2-term complexes over projectives.

Semi-derived Ringel-Hall algebras and Hall algebras of odd-periodic relative derived categories

Let t be a positive integer and A be a hereditary abelian category satisfying some finiteness conditions.We define the semi-derived Ringel-Hall algebra of A from the category C_(Z/t)(A)of Z/t-graded complexes and obtain a natural basis of the semi-derived Ringel-Hall algebra.Moreover,we describe the semiderived Ringel-Hall algebra by the generators and defining relations.In particular,if t is an odd integer,we show an embedding of the derived Hall algebra of the odd-periodic relative derived category in the extended semi-derived Ringel-Hall algebra.