Brauer algebras of type B

For each n=1, we define an algebra having many properties that one might expect to hold for a Brauer algebra of type B_n. It is defined by means of a presentation by generators and relations. We show that this algebra is a subalgebra of the Brauer algebra of type D_{n+1} and point out a cellular structure in it. This work is a natural sequel to the introduction of Brauer algebras of type C_n, which are subalgebras of classical Brauer algebras of type A_{2n-1} and differ from the current ones for n>2. A novel feature is the failure of admissible root sets to describe the tops and bottoms of the diagrams corresponding to monomials in the Brauer algebra of type B_n; instead of these sets we use extended admissible sets in order to find normal forms for monomials in the algebra. Keywords: Brauer algebra; Coxeter Group; diagram representation; root system; Hecke algebra