On Constructing Ideals of the Hall Algebra of Type B

Let H(v)(A(n)) and H(v)(B(n)) be the Hall algebras over Q(v) of the Dynkin quivers A(n) and B(n) (n >= 1), respectively, where nu is an indeterminate and the quivers have linear orientation. By comparing the quantum Serre relations, we find a natural algebra epimorphism pi : H(v)(B(n)) -> H(v2)(A(n)). We determine the kernel of pi by giving two sets of generators. Let phi be the natural algebra homomorphism from H(v)(A(n)) to the quantized Schur algebra S(v)(n + 1, r) (r >= 1) and write (phi) over tilde : H(v2)(A(n)) -> S(v2)(n + 1, r) for the induced map. We obtain several ideals of H(v)(B(n)) by lifting the kernel of phi to the kernel of the composition map (phi) over tilde circle pi : H(v)(B(n)) -> S(v2)(n + 1, r).