Affine and degenerate affine BMW algebras: Actions on tensor space

TL;DR

This paper explores the actions of affine and degenerate affine BMW algebras on tensor space, connecting them to quantum groups and Lie algebras, and identifies their universal parameters as central elements.

Contribution

It defines how these algebras act on tensor space and links their parameters to higher Casimir elements, advancing understanding of their structure and representation theory.

Findings

Actions of affine and degenerate affine BMW algebras on tensor space established

Universal parameters identified as higher Casimir elements

Connections made to quantum groups and Lie algebras

Abstract

The affine and degenerate affine Birman-Murakami-Wenzl (BMW) algebras arise naturally in the context of Schur-Weyl duality for orthogonal and symplectic quantum groups and Lie algebras, respectively. Cyclotomic BMW algebras, affine and cyclotomic Hecke algebras, and their degenerate versions are quotients. In this paper we explain how the affine and degenerate affine BMW algebras are tantalizers (tensor power centralizer algebras) by defining actions of the affine braid group and the degenerate affine braid algebra on tensor space and showing that, in important cases, these actions induce actions of the affine and degenerate affine BMW algebras. We then exploit the connection to quantum groups and Lie algebras to determine universal parameters for the affine and degenerate affine BMW algebras. Finally, we show that the universal parameters are central elements--the higher Casimir elements for orthogonal and symplectic enveloping algebras and quantum groups.