In mathematics, a Hopf algebra, H, is quasitriangular if there exists an invertible element, R such that

• is the coproduct on H,

  ,

  ,

, are algebra morphisms determined by

R is called the R-matrix.

As a consequence of the properties of quasitriangularity, the R-matrix, R, is a solution of the Yang–Baxter equation (and so a module V of H can be used to determine quasi-invariants of braids, knots and links. One may further show that the antipode S must be a linear isomorphism, and thus S2 is an automorphism. In fact, S2 (cf. Ribbon Hopf algebras).

It is possible to construct a quasitriangular Hopf algebra from a Hopf algebra and its dual, using the Drinfeld quantum double construction.

If the Hopf algebra H is quasitriangular, then the category of modules over H is braided with braiding

1. .

Twisting

The property of being a quasi-triangular Hopf algebra is preserved by twisting and satisfying the cocycle condition

, with the twisted comultiplication, R-matrix and co-unit change according to those defined for the quasi-triangular quasi-Hopf algebra. Such a twist is known as an admissible (or Drinfeld) twist.

See also

• Quasi-triangular quasi-Hopf algebra

• Ribbon Hopf algebra

Notes

References