Ribbon Hopf algebra

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

${\displaystyle \nu ^{2}=uS(u),\;S(\nu )=\nu ,\;\varepsilon (\nu )=1}$

${\displaystyle \Delta (\nu )=({\mathcal {R}}_{21}{\mathcal {R}}_{12})^{-1}(\nu \otimes \nu )}$

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

Here

${\displaystyle A}$ is a vector space

${\displaystyle \nabla }$ is the multiplication map ${\displaystyle \nabla :A\otimes A\rightarrow A}$

${\displaystyle \Delta }$ is the co-product map ${\displaystyle \Delta :A\rightarrow A\otimes A}$

${\displaystyle \eta }$ is the unit operator ${\displaystyle \eta$ :\mathbb {C} \rightarrow A}

${\displaystyle \varepsilon }$ is the co-unit operator ${\displaystyle \varepsilon :A\rightarrow \mathbb {C} }$

${\displaystyle S}$ is the antipode ${\displaystyle S:A\rightarrow A}$

${\displaystyle {\mathcal {R}}}$ is a universal R matrix

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

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