On Constructing Ideals of the Hall Algebra of Type B

Abstract

Let Hv(An) and Hv(Bn) be the Hall algebras over ℚ(v) of the Dynkin quivers An and Bn (n ≥ 1), respectively, where v is an indeterminate and the quivers have linear orientation. By comparing the quantum Serre relations, we find a natural algebra epimorphism π : Hv(Bn) → Hv2(An). We determine the kernel of π by giving two sets of generators. Let φ be the natural algebra homomorphism from Hv(An) to the quantized Schur algebra Sv(n + 1, r)(r ≥ 1) and write [Formula: see text] for the induced map. We obtain several ideals of Hv(Bn) by lifting the kernel of φ to the kernel of the composition map [Formula: see text].