A Quotient of the Braid Group Related to Pseudosymmetric Braided Categories *

Motivated by the recently introduced concept of a pseudosymmetric braided monoidal category, we define the pseudosymmetric group PSn, as the quotient of the braid group Bn by the relations σiσ −1 i+1 σi = σi+1σ −1 i σi+1, with 1 ≤ i ≤ n − 2. It turns out that PSn is isomorphic to the quotient of Bn by the commutator subgroup [Pn, Pn] of the pure braid group Pn (which amounts to saying that [Pn, Pn] coincides with the normal subgroup of Bn generated by the elements [σ i , σ i+1], with 1 ≤ i ≤ n− 2), and that PSn is a linear group.