Tangles of type $E_n$ and a reducibility criterion for the Cohen-Wales representation of the Artin group of type $E_6$

TL;DR

This paper introduces tangles of type E_n, constructs a BMW algebra representation of type E_6, and provides a reducibility criterion for the Cohen-Wales representation of the Artin group of type E_6, highlighting conditions for non-semisimplicity.

Contribution

It develops the concept of E_n tangles, constructs a BMW algebra representation of type E_6, and establishes a reducibility criterion for the Cohen-Wales representation.

Findings

Constructed a BMW algebra representation of type E_6.

Identified parameter values where the algebra is not semisimple.

Provided a reducibility criterion for the Cohen-Wales representation.

Abstract

We introduce tangles of type $E_n$ and construct a representation of the Birman-Murakami-Wenzl algebra (BMW algebra) of type $E_6$. As a representation of the Artin group of type $E_6$, this representation is equivalent to the faithful Cohen-Wales representation of type $E_6$ that was used to show the linearity of the Artin group of type $E_6$. We find a reducibility criterion for this representation and complex values of the parameters for which the algebra is not semisimple.