A Categorical Model for the Virtual Braid Group

TL;DR

This paper introduces a new categorical model for the virtual braid group using a strict monoidal category generated by specific morphisms, providing insights into its algebraic structure and connections to quantum algebra.

Contribution

It presents a novel categorical interpretation of the virtual braid group based on pure virtual braids and the algebraic Yang-Baxter equation, linking it to quantum algebra and invariants.

Findings

Categorical framework for virtual braid groups

Connections to quantum algebras and Hopf algebras

Enhanced understanding of representations via Yang-Baxter solutions

Abstract

This paper gives a new interpretation of the virtual braid group in terms of a strict monoidal category SC that is freely generated by one object and three morphisms, two of the morphisms corresponding to basic pure virtual braids and one morphism corresponding to a transposition in the symmetric group. The key to this approach is to take pure virtual braids as primary. The generators of the pure virtual braid group are abstract solutions to the algebraic Yang-Baxter equation. This point of view illuminates representations of the virtual braid groups and pure virtual braid groups via solutions to the algebraic Yang-Baxter equation. In this categorical framework, the virtual braid group is a natural group associated with the structure of algebraic braiding. We then point out how the category SC is related to categories associated with quantum algebras and Hopf algebras and with quantum invariants of virtual links.