Quasitriangular Hopf algebra: Difference between revisions
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= u x u^{-1}</math> where <math>u = m (S \otimes 1)R^{21}</math> (cf. [[Ribbon Hopf algebra]]s). |
= u x u^{-1}</math> where <math>u = m (S \otimes 1)R^{21}</math> (cf. [[Ribbon Hopf algebra]]s). |
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It is possible to construct a quasitriangular Hopf algebra from a Hopf algebra and its dual, using the [[Vladimir Drinfeld|Drinfeld]] |
It is possible to construct a quasitriangular Hopf algebra from a Hopf algebra and its dual, using the [[Vladimir Drinfeld|Drinfeld]] quantum double construction. |
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==Twisting== |
==Twisting== |
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:<math> (F \otimes 1) \circ (\Delta \otimes id) F = (1 \otimes F) \circ (id \otimes \Delta) F </math> |
:<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 |
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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==See also== |
==See also== |
Revision as of 19:59, 16 January 2013
This article needs additional citations for verification. (December 2009) |
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
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 Drinfeld 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.
See also
Notes
- ^ Montgomery & Schneider (2002), Template:Google books quote.
References
- Montgomery, Susan; Schneider, Hans-Jürgen (2002). New directions in Hopf algebras. Vol. 43. Cambridge University Press. ISBN 978-0-521-81512-3.