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

Abstract

The cyclotomic Birman-Murakami-Wenzl (BMW) algebras B_n^k, introduced by R. Häring-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.

45 pages

References in corpus (6)

• The Cyclotomic Birman-Murakami-Wenzl Algebras
• A basis for the Birman-Wenzl algebra
• Specht modules and semisimplicity criteria for Brauer and Birman--Murakami--Wenzl Algebras
• The representation theory of cyclotomic BMW algebras
• The representations of cyclotomic BMW algebras
• The cyclotomic BMW algebra associated with the two string type B braid group

Cited by in corpus (6)

• Remarks on Cyclotomic and Degenerate Cyclotomic BMW Algebras
• Affine and degenerate affine BMW algebras: The center
• Birman-Murakami-Wenzl algebras for general Coxeter groups
• Affine cellularity of affine Birman-Murakami-Wenzl algebras
• Comparison of Admissibility Conditions for Cyclotomic Birman--Wenzl--Murakami Algebras
• Fusion procedure for cyclotomic BMW algebras