[Paper Review] On the freeness of the cyclotomic BMW algebras: admissibility and an isomorphism with the cyclotomic Kauffman tangle algebras

Stewart Wilcox, Sh Shona Yu|ArXiv.org|Nov 27, 2009

Algebraic structures and combinatorial models31 references20 citations

TL;DR

This paper proves that cyclotomic BMW algebras are free R-modules of rank $k^n(2n-1)!!$ over admissible parameter rings, establishing a geometric isomorphism with cyclotomic Kauffman tangle algebras via diagrammatic bases. The proof relies on admissibility conditions derived from representation theory of $\mathscr{B}_2^k$, and constructs explicit algebraic and diagrammatic bases using Jucy-Murphy-type generators and tangle invariants.

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.

Motivation & Objective

• To establish the freeness of cyclotomic BMW algebras $\mathscr{B}_n^k$ over rings with admissible parameters.
• To provide a geometric realization of $\mathscr{B}_n^k$ as a cyclotomic analogue of the Kauffman tangle algebra.
• To construct explicit bases for $\mathscr{B}_n^k$ that are both algebraic and diagrammatic, using tangle invariants.
• To prove that the cyclotomic BMW algebra is isomorphic to the cyclotomic Kauffman tangle algebra $\mathbb{KT}_n^k$ via an explicit isomorphism $\psi$.
• To show that the freeness result holds precisely under admissibility conditions, which are necessary and derived from the representation theory of $\mathscr{B}_2^k$.

Proposed method

• Define the cyclotomic BMW algebra $\mathscr{B}_n^k(R)$ as a quotient of the affine BMW algebra with a $k$-th order relation on the generator $Y$.
• Introduce admissibility conditions on the parameters $A_0, \dots, A_{k-1}, q, \lambda$ to ensure the algebra is well-behaved and free.
• Construct a diagrammatic model of the algebra using affine $n$-tangles in the solid torus, modulo Kauffman skein relations.
• Define a map $\psi: \mathscr{B}_n^k(R) \to \mathbb{KT}_n^k(R)$ from the algebra to the cyclotomic Kauffman tangle algebra, using tangle diagrams and skein relations.
• Prove that $\psi$ is an isomorphism by showing it is injective and surjective, using determinant arguments on the Gram matrix of a trace form.
• Lift bases from a universal ring $R_0$ to arbitrary admissible rings $R$ via base change, preserving freeness and basis structure.

Experimental results

Research questions

• RQ1 Under what conditions on the parameters is the cyclotomic BMW algebra $\mathscr{B}_n^k(R)$ free as an $R$-module?
• RQ2 Can the cyclotomic BMW algebra be geometrically realized as a quotient of a tangle algebra in the solid torus?
• RQ3 Is there an explicit isomorphism between the cyclotomic BMW algebra and a cyclotomic version of the Kauffman tangle algebra?
• RQ4 What is the structure of a basis for $\mathscr{B}_n^k(R)$, both algebraically and diagrammatically?
• RQ5 How do the admissibility conditions on parameters arise from the representation theory of $\mathscr{B}_2^k$?

Key findings

• The cyclotomic BMW algebra $\mathscr{B}_n^k(R)$ is free of rank $k^n(2n-1)!!$ over any admissible parameter ring $R$.
• An isomorphism $\psi: \mathscr{B}_n^k(R) \to \mathbb{KT}_n^k(R)$ is constructed, proving that $\mathscr{B}_n^k(R)$ is isomorphic to the cyclotomic Kauffman tangle algebra.
• The image of the basis $\mathbb{B}_R$ under $\psi$ forms a basis for $\mathbb{KT}_n^k(R)$, confirming the geometric realization.
• The admissibility conditions are necessary and sufficient for freeness and are derived from the representation theory of $\mathscr{B}_2^k$.
• The isomorphism $\psi$ implies that $\mathscr{B}_n^k(R)$ contains $\mathscr{B}_{n-1}^k(R)$ as a subalgebra, via the tangle model.
• The basis elements are described algebraically via products of $\alpha$-chains and $\chi$-elements, and diagrammatically as tangle diagrams with dangles and horizontal arcs.