On Universal Virtual and Welded Braid Groups and Their Linear Representations

TL;DR

This paper introduces and analyzes linear representations of universal virtual and welded braid groups, classifying their local representations and exploring their algebraic properties and quotients.

Contribution

It unifies various braid-type groups through the universal virtual braid group and classifies their local representations, revealing new algebraic structures and quotients.

Findings

Classified complex homogeneous 2- and 3-local representations of UV_n(c).

Established properties of the universal welded braid group, including abelianization and quotients.

Identified three families of 2-local representations of UW_n(c).

Abstract

We introduce linear representations of the universal virtual braid group $UV_n(c)$, where $n\geq 2$ and $c\geq 1$, which is a unifying framework for braid-type groups with multiple types of crossings. We classify and study its complex homogeneous $2$-local representations for all $n\geq 3$ and $c\geq 1$ (unique up to equivalence) and complex homogeneous $3$-local representations for all $n\geq 4$ and $c=2$ (four distinct families). We then introduce the universal welded braid group $UW_n(c)$ as a quotient of $UV_n(c)$ by the welded relations. This group recovers all known welded-type groups as quotients. We prove that $UW_n(c)$ has abelianization $\mathbb{Z}^c \oplus \mathbb{Z}_2$, perfect commutator subgroup for $n \geq 5$, trivial center, and $S_n$ as its smallest non-abelian finite quotient. Finally, we classify and study the complex homogeneous $2$-local representations of $UW_n(c)$ for all $n\geq 3$ and $c\geq 1$, obtaining three distinct families.