Ribbon Hopf algebra

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

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

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

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

$\varepsilon$ 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 $\mathbb {C}$

If $A$ is finite-dimensional, one could equivalently call it ribbon Hopf if and only if its category of (say, left) modules is ribbon; if $A$ is finite-dimensional and quasi-triangular, then it is ribbon if and only if its category of (say, left) modules is pivotal.

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 (1): 83–107. arXiv:hep-th/9202047. Bibcode:1992CMaPh.150...83A. 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.