Ribbon Hopf algebra

A ribbon Hopf algebra (A,∇,η,Δ,ε,S,ℛ,ν) is a quasitriangular Hopf algebra which possess an invertible central element ν more commonly known as the ribbon element, such that the following conditions hold:

ν2=uS(u),S(ν)=ν,ε(ν)=1
Δ(ν)=(ℛ21ℛ12)−1(ν⊗ν)

where u=∇(S⊗id)(ℛ21). 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),ε(uS(u))=1,Δ(uS(u))=(ℛ21ℛ12)−2(uS(u)⊗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
∇ is the multiplication map ∇:A⊗A->A
Δ is the co-product map Δ:A->A⊗A
η is the unit operator η:ℂ->A
ε is the co-unit operator ε:A->ℂ
S is the antipode S:A->A
ℛ is a universal R matrix

We assume that the underlying field K is 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. 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.

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