PBW Basis of the Modified Ringel–Hall Algebra of Type A n via Gröbner–Shirshov Basis

Abstract

In the Ringel–Hall algebra of Dynkin type, the set of all commutator relations between the isoclasses of indecomposable representations forms a minimal Gröbner–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 Gröbner–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 $A_n$ .