On representations of the multi-virtual braid group $M_kVB_n$ and the multi-welded braid group $M_kWB_n$

TL;DR

This paper classifies certain 2-local complex representations of multi-virtual and multi-welded braid groups, revealing their types, unfaithfulness, and constructing new non-local representations extending known braid group representations.

Contribution

It provides a complete classification of 2-local representations of $M_kVB_n$ and $M_kWB_n$, and constructs a new non-local representation extending the LKB representation.

Findings

Exactly $2^{k+1}+1$ types of 2-local representations for $M_kVB_n$

Identifies $3 imes 2^{k-1}+1$ 2-local representations for $M_kWB_n$

Constructs a non-local extension of the LKB representation for $M_2WB_3$

Abstract

This paper classifies complex homogeneous $2$-local representations of the multiple virtual braid group $M_kVB_n$ into $\mathrm{GL}_n(\mathbb{C})$ for $n\geq3$ and $k >1$, showing that such representations fall into exactly $2^{k+1}+1$ distinct types, out of which except three all are unfaithful. In addition, this paper investigates complex homogeneous $2$-local representations of the multiple welded braid group $M_kWB_n$ into $\mathrm{GL}_n(\mathbb{C})$ for $n\geq3$ and $k >1$, identifying $3 \cdot 2^{k-1} +1$ representations. Moreover, the article includes a construction of a non-local representation of $M_2WB_3$ that extends the known LKB representation of the braid group on $3$ strands, namely $B_3$, making a path towards constructing non-local representations of $M_kWB_n$ in general.