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. Haring- 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 as a cyclotomic version of the Kauffman tangle algebra, in terms of affinen-tangles in the solid torus, and produce explicit bases that may be described both algebraically and diagrammatically.