On Universal Virtual and Welded Braid Groups and Their Linear Representations

Mohamad N. Nasser; Oscar Ocampo

paper

On Universal Virtual and Welded Braid Groups and Their Linear Representations

arXiv:2604.19307

Abstract

We introduce linear representations of the universal virtual braid group $U V_{n} (c)$ , 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 $U W_{n} (c)$ as a quotient of $U V_{n} (c)$ by the welded relations. This group recovers all known welded-type groups as quotients. We prove that $U W_{n} (c)$ has abelianization $Z^{c} \oplus Z_{2}$ , 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 $U W_{n} (c)$ for all $n \geq 3$ and $c \geq 1$ , obtaining three distinct families.