Pentagon equations, Delaunay triangulations and pure braid group invariant

We construct $(2n+1)\times (2n+1)$ matrices corresponding to a motion of points on the plane from the point of view of Delaunay triangulations. We define a homomorphism from the pure braid group on ($n+3$) strands to the general linear group ${\text{GL}}_{2n+1}(\mathbb{Q})$.