The center of the BMW algebras and an Okounkov-Vershik like approach

TL;DR

This paper characterizes the center of BMW algebras using Jucys-Murphy elements, introduces a new Okounkov-Vershik like approach to their representations, and explores the structure in both semisimple and non-semisimple cases.

Contribution

It identifies the center of BMW algebras as Wheel Laurent polynomials and develops an Okounkov-Vershik like framework for their finite-dimensional representations.

Findings

Center is given by Wheel Laurent polynomials in semisimple case

Approach separates blocks in non-semisimple case related to type B Lie algebras

Provides new tools for analyzing BMW algebra representations

Abstract

We use the Jucys-Murphy elements of the BMW algebra to show that its center over the complex numbers for almost all parameters making it semisimple is given by Wheel Laurent polynomials, a subalgebra of the symmetric Laurent polynomials in the JM elements. As an application, we give an Okounkov-Vershik like approach to its finite dimensional representations. In the non semisimple case related to the type B Lie algebras, the central subalgebra of Wheel Laurent polynomials is large enough to separate blocks of the BMW algebras.