Braids inside the Birman-Wenzl-Murakami algebra

TL;DR

This paper characterizes the Zariski closure of braid group representations factoring through the Birman-Wenzl-Murakami algebra, establishing unitarizability for certain parameter values and describing the topological closure of their images.

Contribution

It determines the Zariski closure of these representations and proves unitarizability for parameters near 1 with modulus 1, advancing understanding of their algebraic and topological properties.

Findings

Zariski closure of braid group representations is characterized.

Representations are unitarizable for parameters close to 1 on the unit circle.

Topological closure of the image is described under algebraic independence of parameters.

Abstract

We determine the Zariski closure of the representations of the braid groups that factorize through the Birman-Wenzl-Murakami algebra, for generic values of the parameters $\alpha,s$. For $\alpha,s$ of modulus 1 and close to 1, we prove that these representations are unitarizable, thus deducing the topological closure of the image when in addition $\alpha,s$ are algebraically independent.