On representations of the multi-virtual braid group $M_{k}V{B}_n$ and the multi-welded braid group $M_{k}W{B}_n$

Abstract

This paper classifies complex homogeneous $2$-local representations of the multiple virtual braid group $M_{k}V{B}_n$ into ${\mathrm{GL}}_n(\mathbb{C})$ for $n\ge 3$ 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 $M_{k}W{B}_n$ into ${\mathrm{GL}}_n(\mathbb{C})$ for $n\ge 3$ and $k>1$, identifying $3\cdot 2^{k-1}+1$ representations. Moreover, the article includes a construction of a non-local representation of $M_{2}W{B}_3$ 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 $M_{k}W{B}_n$ in general.