PBW Basis of the Modified Ringel-Hall Algebra of Type An via Grobner-Shirshov Basis

In the Ringel-Hall algebra of Dynkin type, the set of all commutator relations between the isoclasses of indecomposable representations forms a minimal Grobner-Shirshov basis, and the set of the corresponding irreducible elements forms a PBW-type basis of the Ringel-Hall algebra. The aim of this paper is to generalize this result to the modified Ringel-Hall algebra of type. First, we compute a minimal Grobner-Shirshov basis for the modified Ringel-Hall algebra of type A(n) by proving all possible compositions between the commutator relations are trivial. Then, by taking the corresponding irreducible monomials, we construct a PBW-type basis for the modified Ringel-Hall algebra of type A(n).