PBW-Basis of Reduced Drinfeld Double 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. We aim to generalize this result to the reduced Drinfeld double Hall algebra of type A(n) . First, we compute a minimal Grobner-Shirshov basis for the reduced Drinfeld double Hall algebra of type An by proving that all possible compositions between the commutator relations are trivial. Then, by taking the corresponding irreducible monomials, we construct a PBW-type basis for the reduced Drinfeld double Hall algebra of type A(n) .