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:

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.

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