Ribbon Hopf algebra

A Ribbon Hopf algebra $(A,m,\Delta ,u,\varepsilon ,\Phi ,{\mathcal {R}},\nu )$ is a Quasi-triangular Hopf algebra which possess an invertible central element $\nu$ more commonly known as the ribbon element, such that the following conditions hold:

$\nu ^{2}=uS(u),\;S(\nu )=\nu ,\;\varepsilon (\nu )=1$

$\Delta (\nu )=({\mathcal {R}}_{21}{\mathcal {R}}_{12})^{-1}(\nu \otimes \nu )$

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

Where

$A$ 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 $u:A\rightarrow \mathbb {C}$

$\varepsilon$ is the co-unit opertor $\varepsilon :\mathbb {C} \rightarrow A$

$\Phi$ is the co-associator isomorphism such that $\Phi _{123}=\Phi =a_{1}\otimes a_{2}\otimes a_{3}\in A\otimes A\otimes A$.

${\mathcal {R}}$ is a universal R matrix

See also

• Quasitriangular Hopf algebra
• Quasi-triangular Quasi-Hopf algebra

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