Ribbon Hopf algebra

A ribbon Hopf algebra $(A,m,\Delta,u,\varepsilon,S,\mathcal{R},\nu)$ 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 $u=m(S\otimes \text{id})(\mathcal{R}_{21})$. 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

$m$ is the multiplication map $m:A \otimes A \rightarrow A$

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

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

$\varepsilon$ 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 $\mathbb{C}$

See also

• Quasitriangular Hopf algebra
• Quasi-triangular quasi-Hopf algebra

References

• Altschuler, D.; Coste, A. (1992). "Quasi-quantum groups, knots, three-manifolds and topological field theory". Commun. Math. Phys. 150: 83–107. arXiv:hep-th/9202047. doi:10.1007/bf02096567.
• Chari, V. C.; Pressley, A. (1994). A Guide to Quantum Groups. Cambridge University Press. ISBN 0-521-55884-0.
• Drinfeld, Vladimir (1989). "Quasi-Hopf algebras". Leningrad Math J. 1: 1419–1457.
• Majid, Shahn (1995). Foundations of Quantum Group Theory. Cambridge University Press.