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 co-product map is the unit operator is the co-unit operator is the antipode is a universal R matrix

We assume that the underlying field is