Cyclotomic Birman–Wenzl–Murakami Algebras, II: Admissibility Relations and Freeness

Abstract

The cyclotomic Birman-Wenzl-Murakami algebras are quotients of the affine BMW algebras in which the affine generator satisfies a polynomial relation. We study admissibility conditions on the ground ring for these algebras, and show that the algebras defined over an admissible integral ground ring S are free S-modules and isomorphic to cyclotomic Kauffman tangle algebras. We also determine the representation theory in the generic semisimple case, obtain a recursive formula for the weights of the Markov trace, and give a sufficient condition for semisimplicity.

Non-vanishing Gram determinants for cyclotomic Nazarov–Wenzl and Birman–Murakami–Wenzl algebras

“…However, he did assume that ω 0 is invertible. Recently, B r,n have been studying extensively by three groups of mathematicians in [12,17,18,13,36,33,41,[38][39][40]42], etc.…”

“…We thank Professor F. Goodman for his comments on the presentation of the paper. been studied extensively by Goodman and Hauschild, Wilcox and Yu, Xu and the authors in [12,17,13,18,33,36,[38][39][40][41][42]. In particular, it has been proved in [38] that the cyclotomic BMW algebras are cellular algebras in the sense of [19].…”

The Representations of Cyclotomic BMW Algebras, II

“…In [8], Goodman and Hauchild Moseley introduced the notion of strong u-admissible conditions. In [22], Yu introduced the notion of "admissible"-conditions.…”

Cellularity of cyclotomic Birman–Wenzl–Murakami algebras

“…(See [6,7,19,21].) If S is an admissible integral domain, then the assignment e i → E i , g i → G i , y 1 → ρX i determines an isomorphism of W n,S,r and KT n,S,r .…”

“…It has been shown in [6,7,19,21] that if S is an admissible integral domain, then W n,S,r is a free S-module of rank r n (2n − 1)! !, and is isomorphic to a cyclotomic version of the Kauffman tangle algebra.…”