Quasitriangular Hopf algebra: Difference between revisions
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It is possible to contruct a quasitriangular Hopf algebra from a Hopf algebra and its dual, using the Drinfrl'd [[quantum double]] construction. |
It is possible to contruct a quasitriangular Hopf algebra from a Hopf algebra and its dual, using the Drinfrl'd [[quantum double]] construction. |
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== Twisting == |
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The property of being a [[quasi-triangular Hopf algebra]] is preserved by [[Quasi-bialgebra#Twisting|twisting]] via an invertible element <math> F = \sum_i f^i \otimes f_i \in \mathcal{A \otimes A} </math> such that <math> (\varepsilon \otimes id )F = (id \otimes \varepsilon)F = 1 </math> and satisfying the cocycle condition |
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:<math> (F \otimes 1) \circ (\Delta \otimes id) F = (1 \otimes F) \circ (id \otimes \Delta) F </math> |
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Furthermore, <math> u = \sum_i f^i S(f_i)</math> is invertible and the twisted antipode is given by <math>S'(a) = u S(a)u^{-1}</math>, 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. |
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[[Category:Hopf algebras]] |
[[Category:Hopf algebras]] |
Revision as of 06:19, 1 May 2006
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 morphisns 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, , and , and so .
It is possible to contruct a quasitriangular Hopf algebra from a Hopf algebra and its dual, using the Drinfrl'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
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.