Ribbon Hopf algebra

A ribbon Hopf algebra [math]\displaystyle{ (A,\nabla, \eta,\Delta,\varepsilon,S,\mathcal{R},\nu) }[/math] is a quasitriangular Hopf algebra which possess an invertible central element [math]\displaystyle{ \nu }[/math] more commonly known as the ribbon element, such that the following conditions hold:

[math]\displaystyle{ \nu^{2}=uS(u), \; S(\nu)=\nu, \; \varepsilon (\nu)=1 }[/math]

[math]\displaystyle{ \Delta (\nu)=(\mathcal{R}_{21}\mathcal{R}_{12})^{-1}(\nu \otimes \nu ) }[/math]

where [math]\displaystyle{ u=\nabla(S\otimes \text{id})(\mathcal{R}_{21}) }[/math]. Note that the element u exists for any quasitriangular Hopf algebra, and [math]\displaystyle{ uS(u) }[/math] must always be central and satisfies [math]\displaystyle{ S(uS(u))=uS(u), \varepsilon(uS(u))=1, \Delta(uS(u)) = (\mathcal{R}_{21}\mathcal{R}_{12})^{-2}(uS(u) \otimes uS(u)) }[/math], so that all that is required is that it have a central square root with the above properties.

Here

[math]\displaystyle{ A }[/math] is a vector space

[math]\displaystyle{ \nabla }[/math] is the multiplication map [math]\displaystyle{ \nabla:A \otimes A \rightarrow A }[/math]

[math]\displaystyle{ \Delta }[/math] is the co-product map [math]\displaystyle{ \Delta: A \rightarrow A \otimes A }[/math]

[math]\displaystyle{ \eta }[/math] is the unit operator [math]\displaystyle{ \eta:\mathbb{C} \rightarrow A }[/math]

[math]\displaystyle{ \varepsilon }[/math] is the co-unit operator [math]\displaystyle{ \varepsilon: A \rightarrow \mathbb{C} }[/math]

[math]\displaystyle{ S }[/math] is the antipode [math]\displaystyle{ S: A\rightarrow A }[/math]

[math]\displaystyle{ \mathcal{R} }[/math] is a universal R matrix

We assume that the underlying field [math]\displaystyle{ K }[/math] is [math]\displaystyle{ \mathbb{C} }[/math]

If [math]\displaystyle{ A }[/math] is finite-dimensional, one could equivalently call it ribbon Hopf if and only if its category of (say, left) modules is ribbon; if [math]\displaystyle{ A }[/math] 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. doi:10.1007/bf02096567. Bibcode: 1992CMaPh.150...83A. 
• Chari, V. C.; Pressley, A. (1994). A Guide to Quantum Groups. Cambridge University Press. ISBN 0-521-55884-0. https://archive.org/details/guidetoquantumgr0000char. 
• Drinfeld, Vladimir (1989). "Quasi-Hopf algebras". Leningrad Math J. 1: 1419–1457. 
• Majid, Shahn (1995). Foundations of Quantum Group Theory. Cambridge University Press.