Tangles of type E_n and a reducibility criterion for the Cohen-Wales representation of the Artin group of type E₆

2011/06/22 by Claire Levaillant, Claire I. Levaillant, Levaillant, Claire I.
Mathematics · #20F36 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT) #Rings and Algebras (math.RA) #math.GR #math.RA #math.RT #msc:20F36

paper · pdf · doi:10.48550/arxiv.1106.4463

34 pages, 17 figures

arxiv created 2011/06/22 · openalex publication_date 2011/06/22 · arxiv updated 2011/06/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We introduce tangles of type E_n and construct a representation of the Birman-Murakami-Wenzl algebra (BMW algebra) of type E₆. As a representation of the Artin group of type E₆, this representation is equivalent to the faithful Cohen-Wales representation of type E₆ that was used to show the linearity of the Artin group of type E₆. We find a reducibility criterion for this representation and complex values of the parameters for which the algebra is not semisimple.

Citations

• Classification of the invariant subspaces of the Lawrence-Krammer representation
• Classification of the invariant subspaces of the Cohen-Wales representation of the Artin group of type D_n
• A basis for the Birman-Wenzl algebra
• The Burman-Murakami-Wenzl algebras of type En

Related

• Reducibility of the Cohen-Wales representation of the Artin group of type D_n
• Irreducibility of the Lawrence-Krammer representation of the BMW algebra of type A_n-1
• BMW algebras of simply laced type
• Irreducibility of the Lawrence-Krammer representation of the BMW algebra of type A_n-1, PhD thesis California Institute of Technology 2008
• Dynamical systems and operator algebras associated to Artin's representation of braid groups