Q-centralizer algebras for spin groups

Abstract

Let V and V ϵ be the vector and a spinor representation of so N , N∈ N . We show that the quantum group U q ( so N ) and the braid group corresponding to the Dynkin diagram B f (the latter acting via R -matrices) are each others commutant on V ϵ ⊗ V ⊗ f . Moreover, these braid representations factor through specializations of Häring–Oldenburg's B-BMW-algebra; this also holds with V ϵ replaced by V mϵ , m∈ N , if N is even. We use this observation to compute the weights of the Markov trace of this algebra as a 2-variable function, and the values of the parameters for which it is semisimple.