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

Generalized virtual braid groups, quasi-shuffle product and quantum groups