# Ribbon Hopf algebra

A ribbon Hopf algebra ${\displaystyle (A,m,\Delta ,u,\varepsilon ,S,{\mathcal {R}},\nu )}$ 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 ${\displaystyle u=m(S\otimes {\text{id}})({\mathcal {R}}_{21})}$. 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
${\displaystyle m}$ is the multiplication map ${\displaystyle m:A\otimes A\rightarrow A}$
${\displaystyle \Delta }$ is the co-product map ${\displaystyle \Delta :A\rightarrow A\otimes A}$
${\displaystyle u}$ is the unit operator ${\displaystyle u:\mathbb {C} \rightarrow A}$
${\displaystyle \varepsilon }$ 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 ${\displaystyle \mathbb {C} }$