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 )

where $u=m(S\otimes \text{id})(\mathcal{R}_{21})$ . 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

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 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 $\mathbb{C}$

References

Altschuler, D.; Coste, A. (1992). "Quasi-quantum groups, knots, three-manifolds and topological field theory". Commun. Math. Phys. 150: 83–107. arXiv:hep-th/9202047. 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.

Ribbon Hopf algebra

See also

References