Generalized Virtual Braid Groups, Quasi-Shuffle Product and Quantum Groups

Abstract

We introduce in this notice the generalized virtual braid group on n strands GVBn, generalizing simultaneously the braid groups and their virtual versions. A Matsumoto–Tits-type section lifting shuffles in a symmetric group |$\mathfrak {S}_n$| to the monoid associated to GVBn is constructed, which is then applied to characterize the quantum quasi-shuffle product. A family of representations of GVBn is constructed using quantum groups.