Ribbon Hopf algebra

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A Ribbon Hopf algebra (A,m,\Delta,u,\varepsilon,S,\mathcal{R},\nu) 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:

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

such that u=m(S\otimes 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
Δ 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 opertor \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}

[edit] See also

[edit] 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