Ribbon Hopf algebra

A ribbon Hopf algebra $$(A,\nabla, \eta,\Delta,\varepsilon,S,\mathcal{R},\nu)$$ is a quasitriangular Hopf algebra which possess an invertible central element $$\nu$$ more commonly known as the ribbon element, such that the following conditions hold:


 * $$\nu^{2}=uS(u), \; S(\nu)=\nu, \; \varepsilon (\nu)=1$$
 * $$\Delta (\nu)=(\mathcal{R}_{21}\mathcal{R}_{12})^{-1}(\nu \otimes \nu )$$

where $$u=\nabla(S\otimes \text{id})(\mathcal{R}_{21})$$. Note that the element u exists for any quasitriangular Hopf algebra, and $$uS(u)$$ must always be central and satisfies $$S(uS(u))=uS(u), \varepsilon(uS(u))=1, \Delta(uS(u)) = (\mathcal{R}_{21}\mathcal{R}_{12})^{-2}(uS(u) \otimes uS(u))$$, so that all that is required is that it have a central square root with the above properties.

Here
 * $$ A $$ is a vector space
 * $$ \nabla $$ is the multiplication map $$\nabla:A \otimes A \rightarrow A$$
 * $$ \Delta $$ is the co-product map $$\Delta: A \rightarrow A \otimes A$$
 * $$ \eta $$ is the unit operator $$\eta:\mathbb{C} \rightarrow A$$
 * $$ \varepsilon $$ is the co-unit operator $$\varepsilon: A \rightarrow \mathbb{C}$$
 * $$ S $$ is the antipode $$S: A\rightarrow A$$
 * $$\mathcal{R}$$ is a universal R matrix

We assume that the underlying field $$K$$ is $$\mathbb{C}$$

If $$ A $$ is finite-dimensional, one could equivalently call it ribbon Hopf if and only if its category of (say, left) modules is ribbon; if $$ A $$ is finite-dimensional and quasi-triangular, then it is ribbon if and only if its category of (say, left) modules is pivotal.