A quotient of the braid group related to pseudosymmetric braided categories

TL;DR

This paper introduces the pseudosymmetric group PS_n as a quotient of the braid group, explores its algebraic properties, and establishes its isomorphism with a certain quotient related to pure braid groups, highlighting its linearity.

Contribution

It defines the pseudosymmetric group PS_n via specific relations, proves its isomorphism with a quotient of the braid group, and demonstrates its linearity, connecting it to pseudosymmetric braided categories.

Findings

PS_n is isomorphic to the quotient of B_n by the commutator subgroup of P_n.

PS_n is a linear group.

The relations defining PS_n relate to pseudosymmetric braided monoidal categories.

Abstract

Motivated by the recently introduced concept of a pseudosymmetric braided monoidal category, we define the pseudosymmetric group PS_n, as the quotient of the braid group B_n by the relations \sigma_i\sigma_{i+1}^{-1}\sigma_i=\sigma _{i+1}\sigma_i^{-1}\sigma_{i+1}, with 1\leq i\leq 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 [\sigma_i^2, \sigma_{i+1}^2], with 1\leq i\leq n-2), and that PS_n is a linear group.