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
![{\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,
; 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.
If the Hopf algebra H is quasitriangular, then the category of modules over H is braided with braiding
.
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)\cdot (\Delta \otimes id)(F)=(1\otimes F)\cdot (id\otimes \Delta )(F)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/958ce7cc2353fe4a95ebfffb0e7834eac72308ee)
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
- Quasi-triangular quasi-Hopf algebra
- Ribbon Hopf algebra
Notes
- ^ Montgomery & Schneider (2002), p. 72.
References
- Montgomery, Susan (1993). Hopf algebras and their actions on rings. Regional Conference Series in Mathematics. Vol. 82. Providence, RI: American Mathematical Society. ISBN 0-8218-0738-2. Zbl 0793.16029.
- Montgomery, Susan; Schneider, Hans-Jürgen (2002). New directions in Hopf algebras. Mathematical Sciences Research Institute Publications. Vol. 43. Cambridge University Press. ISBN 978-0-521-81512-3. Zbl 0990.00022.