Quasitriangular Hopf algebra: Difference between revisions
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In [[mathematics]], a [[Hopf algebra]], ''H'', is '''quasitriangular''' if there exists an [[inverse element|invertible]] element, ''R'', of <math>H \otimes H</math> such that |
In [[mathematics]], a [[Hopf algebra]], ''H'', is '''quasitriangular'''<ref>Montgomery & Schneider (2002), {{Google books quote|id=I3IK9U5Co_0C|page=72|text=Quasitriangular|p. 72}}.</ref> if there exists an [[inverse element|invertible]] element, ''R'', of <math>H \otimes H</math> such that |
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:*<math>R \ \Delta(x) = (T \circ \Delta)(x) \ R</math> for all <math>x \in H</math>, where <math>\Delta</math> is the coproduct on ''H'', and the linear map <math>T : H \otimes H \to H \otimes H</math> is given by <math>T(x \otimes y) = y \otimes x</math>, |
:*<math>R \ \Delta(x) = (T \circ \Delta)(x) \ R</math> for all <math>x \in H</math>, where <math>\Delta</math> is the coproduct on ''H'', and the linear map <math>T : H \otimes H \to H \otimes H</math> is given by <math>T(x \otimes y) = y \otimes x</math>, |
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* [[Quasi-triangular Quasi-Hopf algebra]] |
* [[Quasi-triangular Quasi-Hopf algebra]] |
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* [[Ribbon Hopf algebra]] |
* [[Ribbon Hopf algebra]] |
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== Notes == |
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<references/> |
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== References == |
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* [[Susan Montgomery]], [[Hans-Jürgen Schneider]]. ''New directions in Hopf algebras'', Volume 43. Cambridge University Press, 2002. ISBN 9780521815123 |
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Revision as of 14:33, 6 May 2010
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 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
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
Notes
- ^ Montgomery & Schneider (2002), Template:Google books quote.
References
- Susan Montgomery, Hans-Jürgen Schneider. New directions in Hopf algebras, Volume 43. Cambridge University Press, 2002. ISBN 9780521815123