Ribbon Hopf algebra

Definition
A quasitriangular Hopf algebra $$H$$ with multiplication $$\nabla$$, comultiplication $$\Delta$$, unit $$\eta$$, counit $$\epsilon$$, antipode map $$S$$ and R-matrix $$R$$, is said to be a ribbon Hopf algebra if it has an element $$\nu$$ (called a ribbon element) such that:


 * $$\nu$$ is an invertible central element of $$H$$
 * $$\nu^2 = \eta S(\eta)$$
 * $$S(\nu) = \nu$$
 * $$\epsilon(\nu) = 1$$
 * $$\Delta(\nu) = (R_{21}R_{12})^{-1} (\nu \otimes \nu)$$

Equivalently, $$\nu$$ is a central squareroot of $$\eta S(\eta)$$.

By $$ab$$ we actually mean $$\nabla(a \otimes b)$$.

Weaker properties

 * Quasitriangular Hopf algebra
 * Hopf algebra