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: 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.
Other articles related to "quasitriangular hopf algebra, hopf algebra":
... 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 ...
Famous quotes containing the word algebra:
“Poetry has become the higher algebra of metaphors.”
—José Ortega Y Gasset (18831955)