Quasitriangular Hopf algebra

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In mathematics, a Hopf algebra, H, is quasitriangular[1] if there exists an invertible element, R, of such that

  • for all , where is the coproduct on H, and the linear map is given by ,
  • ,
  • ,

where , , and , where , , and , 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). Also as a consequence of the properties of quasitriangularity, ; moreover , , and . One may further show that the antipode S must be a linear isomorphism, and thus S2 is an automorphism. In fact, S2 is given by conjugating by an invertible element: where (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.


The property of being a quasi-triangular Hopf algebra is preserved by twisting via an invertible element such that and satisfying the cocycle condition

Furthermore, is invertible and the twisted antipode is given by , 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[edit]


  1. ^ Montgomery & Schneider (2002), p. 72.