A Categorical Structure for the Virtual Braid Group

Abstract

This article 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. This point of view makes many relationships between the virtual braid group and the pure virtual braid group apparent, and makes representations of the virtual braid groups and pure virtual braid groups via solutions to the algebraic Yang–Baxter Equation equally transparent. In this categorical framework, the virtual braid group has nothing to do with the plane and nothing to do with virtual crossings. It is a natural group associated with the structure of algebraic braiding.

Key Words:

• Pure virtual braid group
• Pure welded braid group
• Virtual braid group
• Welded braid group

2000 Mathematics Subject Classification:

• 57M27

ACKNOWLEDGMENT

Both authors were partially supported by UIC, NTUA, and MFO.

Notes

Communicated by H.-J. Schneider.

Dedicated to Miriam Cohen on the occasion of her retirement.