A ribbon Hopf algebra is a quasitriangular Hopf algebra which possess an invertible central element more commonly known as the ribbon element, such that the following conditions hold: where . Note that the element u exists for any quasitriangular Hopf algebra, and must always be central and satisfies , so that all that is required is that it have a central square root with the above properties. Here is a vector space is the multiplication map is the coproduct map is the unit operator is the counit operator is the antipode is a universal R matrix We assume that the underlying field is
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