Yang–Baxter equation and representations of the virtual braid group

It is well known that a solution for the Yang–Baxter equation (YBE) or equivalently for the braid equation (BE) gives a representation of the braid group [Formula: see text]. This paper is devoted to the generalization of this result to the case of the virtual braid group, related to the theory of virtual knots and playing an important role in quantum field theories. In this paper we explore a connection between YBE and representations of the virtual braid group [Formula: see text]. In particular, we show that any solution [Formula: see text] for the YBE with invertible [Formula: see text] defines a representation of the virtual pure braid group [Formula: see text], for any [Formula: see text], into [Formula: see text] for linear solution and into [Formula: see text] for set-theoretic solution. Any solution of the BE with invertible [Formula: see text] gives a representation of a normal subgroup [Formula: see text] of [Formula: see text]. As a consequence of these two results we get that any invertible solution for the BE or YBE gives a representation of [Formula: see text]. We also elaborate the technique of the group [Formula: see text] connected with the problem of extension of YBE solutions, that is the construction of the YBE solution on the direct product [Formula: see text] by solutions on the factors [Formula: see text] and [Formula: see text].