The cyclotomic BMW algebra associated with the two string type B braid group

The cyclotomic Birman–Murakami–Wenzl (or BMW) algebras ??kn, introduced by R. Häring–Oldenburg, are extensions of the cyclotomic Hecke algebras of Ariki–Koike, in the same way as the BMW algebras are extensions of the Iwahori–Hecke algebras of type A. In this article, we produce a basis of the two strand algebra ??k2 and construct its left regular representation. In the process we obtain relations amongst the parameters which play an important role in the n strand case.