Braids inside the Birman–Wenzl–Murakami algebra

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 α, s. For α, 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 α, s are algebraically independent.

Discrete real specializations of sesquilinear representations of the braid groups

“…1=˛, where m D ˛C 1=˛. Budney proved that the Lawrence-Krammer representation is sesquilinear [4], and Brunat and Marin give a more general proof that all the BMW representations are sesquilinear [3]; see also [10]. It is also known that the sesquilinear forms J l;m are positive definite for a neighborhood of .1; 1/ in the unit complex sphere in C 2 [17]; see [2, Theorem 1.2] for a concise restatement.…”

Image of the braid groups inside the finite Temperley–Lieb algebras

“…A natural question concerns the image of B n inside its classical linear representations, the most classical ones being the ones which factor through the Hecke algebra H n (α) of type A n−1 , such as the Burau or the Jones representation. Inside an infinite field, the determination of the Zariski closure of such representations in the generic case is completely known by [FLW] and [Mar1] ; actually the more general cases of the representations of the Birman-Wenzl-Murakami algebra and of the Hecke algebras for other reflection groups is also known by [Mar4,Mar2], and more precise information on the Jones representation can be found in [FLW] and [Ku] in the non-generic case.…”