Ribbon Hopf algebra

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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.


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

See also[edit]


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