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.

References