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
![{\displaystyle \phi _{12}(a\otimes b)=a\otimes b\otimes 1,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a86745934a6d24c3f3265b99e9a50fc6f3e1e583)
![{\displaystyle \phi _{13}(a\otimes b)=a\otimes 1\otimes b,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/70ed2de29e791f28c17f968e3728ca5a28551f55)
![{\displaystyle \phi _{23}(a\otimes b)=1\otimes a\otimes b.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/18cb684dcb7c39cf24389e3e89925af624ac8c75)
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,
, and
, and so
.
It is possible to contruct a quasitriangular Hopf algebra from a Hopf algebra and its dual, using the Drinfel'd quantum double construction.
Twisting
The property of being a quasi-triangular Hopf algebra is preserved by twisting via an invertible element
such that
and satisfying the cocycle condition
![{\displaystyle (F\otimes 1)\circ (\Delta \otimes id)F=(1\otimes F)\circ (id\otimes \Delta )F}](https://wikimedia.org/api/rest_v1/media/math/render/svg/00099fc78972f1b05ae6146dd824f060c01fd430)
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 Drinfel'd) twist.
See also