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Ribbon Hopf algebra

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A ribbon Hopf algebra is a quasitriangular Hopf algebra which possess an invertible central element more commonly known as the ribbon element, such that the following conditions hold:

where . Note that the element u exists for any quasitriangular Hopf algebra, and must always be central and satisfies , so that all that is required is that it have a central square root with the above properties.

Here

is a vector space
is the multiplication map
is the co-product map
is the unit operator
is the co-unit operator
is the antipode
is a universal R matrix

We assume that the underlying field is

If is finite-dimensional, one could equivalently call it ribbon Hopf if and only if its category of (say, left) modules is ribbon; if is finite-dimensional and quasi-triangular, then it is ribbon if and only if its category of (say, left) modules is pivotal.

See also

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References

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  • Altschuler, D.; Coste, A. (1992). "Quasi-quantum groups, knots, three-manifolds and topological field theory". Commun. Math. Phys. 150 (1): 83–107. arXiv:hep-th/9202047. Bibcode:1992CMaPh.150...83A. 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.