On $q$-monodromy groups of singularities

In this paper some facts are proved concerning the $q$-analogue of symmetric and skew-symmetric monodromy groups of singularities. The image of the Burau representation of the group of braids on three strings is described. For $q$ a root of 1 the image of the group of $q$-coloured braids of a singularity in the skew-symmetric monodromy group is studied. The latter is distinguished among the $q$-monodromy groups for the roots of 1. The orbits of the braid monodromy group over $\mathbb Z$ are described. A connection is established between the spectra of the generalized Cartan $q$-matrix and the Coxeter $q$-operator for Coxeter–Dynkin diagrams without cycles of odd length.