Quasitriangular Hopf Algebra

Quasitriangular Hopf Algebra

In mathematics, a Hopf algebra, H, is quasitriangular 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 S^2 is an automorphism. In fact, S^2 is given by conjugating by an invertible element: S(x)
= u x u^{-1} where (cf. Ribbon Hopf algebras).

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

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