A quotient of the braid group related to pseudosymmetric braided categories

Florin Panaite and Mihai D. Staic

Vol. 244 (2010), No. 1, 155–167

DOI: 10.2140/pjm.2010.244.155

Abstract

Motivated by the recent concept of a pseudosymmetric braided monoidal category, we define the pseudosymmetric group PS_n to be the quotient of the braid group B_n by the relations σ_iσ_i+1⁻¹σ_i = σ_i+1σ_i⁻¹σ_i+1 with 1 ≤ i ≤ n − 2. It turns out that PS_n is isomorphic to the quotient of B_n by the commutator subgroup [P_n,P_n] of the pure braid group P_n (which amounts to saying that [P_n,P_n] coincides with the normal subgroup of B_n generated by the elements [σ_i²,σ_i+1²] with 1 ≤ i ≤ n− 2), and that PS_n is a linear group.

Keywords

braid group, symmetric group, braided categories, pseudosymmetric braidings

Mathematical Subject Classification 2000

Primary: 20F36

Secondary: 18D10

Milestones

Received: 17 December 2008

Revised: 26 June 2009

Accepted: 2 July 2009

Published: 17 November 2009