Gröbner–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 Gröbner-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 Db(An) using the Auslander-Reiten quiver of Db(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).