Quantum Groups and Quantum Semigroups

We construct a functor from a certain category of quantum semigroups to a category of quantum groups, which, for example, assigns Fun(Matq(N)) to Fun(GLq(N)). Combining with (a generalization of) the Faddeev–Reshetikhin–Takhtadzhyan construction, we obtain quantum groups with universalR-matrices for every constant solution of the Yang–Baxter equation of both vertex and face type.