  We introduce in this paper the generalized virtual braid group on n strands
GVB_n, generalizing simultaneously the braid groups and their virtual versions.
A Mastumoto-Tits type section lifting shuffles in a symmetric group S_n to the
monoid associated to GVB_n is constructed, which is then applied to
characterize the quantum quasi-shuffle product. A family of representations of
GVB_n is constructed using quantum groups.

Quantum Algebra;Combinatorics;Rings and Algebras