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

TL;DR

This paper proves the freeness and rank of cyclotomic BMW algebras under admissibility conditions, and establishes an isomorphism with cyclotomic Kauffman tangle algebras, providing explicit bases and geometric realization.

Contribution

It demonstrates the freeness and explicit basis construction of cyclotomic BMW algebras, and establishes their isomorphism with cyclotomic Kauffman tangle algebras under admissibility conditions.

Findings

Proved the algebra is free of rank k^n (2n-1)!!

Established a geometric realization as affine n-tangles in the solid torus

Provided explicit algebraic and diagrammatic bases

Abstract

The cyclotomic Birman-Murakami-Wenzl (BMW) algebras B_n^k, introduced by R. H\"aring-Oldenburg, are a generalisation of the BMW algebras associated with the cyclotomic Hecke algebras of type G(k,1,n) (aka 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 originally arise from the representation theory of B_2^k, which is analysed by the authors in a previous paper. Furthermore, we obtain a geometric realisation of B_n^k 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. The admissibility conditions are the most general offered in the literature for which these results hold; they are necessary and sufficient for all results for general n.