On the freeness of the cyclotomic BMW algebras : admissibility and an isomorphism with the cyclotomic Kauffman tangle algebras

The cyclotomic Birman-Murakami-Wenzl (or BMW) algebras B^k_n , introduced by R. Häring-Oldenburg, are a generalisation of the BMW algebras associated with the cyclotomic Hecke algebras of type G(k,1,n) (also known as Ariki-Koike algebras) and type B knot theory. In this paper, we prove the algebra is free and of rank k^n(2n - 1)!! over ground rings with parameters satisfying so-called "admissibility conditions". These conditions are necessary in order for these results to hold and arise from the representation theory of B^k_2 , which is analysed by the authors in a previous paper. Furthermore, we obtain a geometric realisation of B^k_n as a cyclotomic version of the Kauffman tangle algebra, in terms of affine n-tangles in the solid torus, and produce explicit bases that may be described both algebraically and diagrammatically. Key words: cyclotomic BMW algebras; cyclotomic Hecke algebras; Ariki-Koike algebras; Brauer algebras; affine tangles; Kauffman link invariant; admissibility