Ribbon Hopf algebra

A Ribbon Hopf algebra $(A,m,Delta,u,varepsilon,S,mathcal\{R\},\; u)$ is a Quasitriangular Hopf algebrawhich possess an invertible central element $u$ more commonly known as the ribbon element, such that the following conditions hold:

:$u^\{2\}=uS(u),\; ;\; S(\; u)=\; u,\; ;\; varepsilon\; (\; u)=1$:$Delta\; (\; u)=(mathcal\{R\}\_\{21\}mathcal\{R\}\_\{12\})^\{-1\}(\; u\; otimes\; u\; )$

such that $u=m(Sotimes\; id)(mathcal\{R\}\_\{21\})$. 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 withthe above properties.

Here:$A$ is a vector space:$m$ is the multiplication map $m:A\; otimes\; A\; ightarrow\; A$:$Delta$ is the co-product map $Delta:\; A\; ightarrow\; A\; otimes\; A$:$u$ is the unit operator $u:mathbb\{C\}\; ightarrow\; A$:$varepsilon$ is the co-unit opertor $varepsilon:\; A\; ightarrow\; mathbb\{C\}$:$S$ is the antipode $S:\; A\; ightarrow\; A$:$mathcal\{R\}$ 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