On the cellularity of the cyclotomic Birman-Murakami-Wenzl algebras

Abstract

The cyclotomic Birman–Murakami–Wenzl (or BMW) algebras ℬ n k were introduced by Häring-Oldenburg as a generalization 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 the PhD thesis of the second author and previous work of the authors, these algebras are shown to be free of rank kn (2n−1)!! over ground rings with parameters satisfying the so-called ‘admissibility conditions’, the most general conditions under which these results can hold. These conditions are build from the representation theory of ℬ 2 k and are necessary and sufficient in order for these results to hold. (A review of these admissibility conditions and their differences with others used in the literature is provided in this paper.) Furthermore, ℬ n k may be topologically realized, in terms of affine/cylindrical tangles, as a cyclotomic version of the Kauffman tangle algebra, and bases that can be explicitly described both algebraically and diagrammatically were obtained. In this paper, it is shown that these bases combined with a carefully constructed lifting of an arbitrary cellular basis of the Ariki–Koike algebras yield a cellular basis of the cyclotomic BMW algebras, in the sense of Graham and Lehrer, thereby establishing cellularity of the algebra under the most general conditions available in the literature.