19.7 (2018)

The group of virtual pure braids $VP_n$, $n \geqslant 2$, is generated by elements $\lambda_{ij}$, $1 \leqslant i \neq j \leqslant n$, and is defined by the relations $\lambda_{ij}\lambda_{kl} = \lambda_{kl}\lambda_{ij}$, $\lambda_{ki}\lambda_{kj}\lambda_{ij} = \lambda_{ij}\lambda_{kj}\lambda_{ki}$, where different letters denote different indices.
$\qquad$ a) Construct a normal form of words in the group $VP_n$ for $n \geqslant 4$.
$\qquad$ b) Is the group $VP_n$ linear for $n \geqslant 4$? It is known that the group $VP_3$ is linear.