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:

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

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

such that $u=m(S\otimes id)({\mathcal {R}}_{21})$. 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 opertor $\varepsilon :A\rightarrow \mathbb {C}$

$S$ is the antipode such that $S\otimes id:A\otimes A\rightarrow A\otimes A$

${\mathcal {R}}$ 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.: Quasi-quantum groups, knots, three-manifolds and topological field theory. Commun. Math. Phys. 150 1992 83-107 http://arxiv.org/pdf/hep-th/9202047
• Chari, V.C., Pressley, A.: A Guide to Quantum Groups Cambridge University Press, 1994 ISBN 0-521-55884-0.
• Vladimir Drinfeld, Quasi-Hopf algebras, Leningrad Math J. 1 (1989), 1419-1457
• Majid, S.: Foundations of Quantum Group Theory Cambridge University Press, 1995