Quasitriangular Hopf algebra: Difference between revisions
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In [[mathematics]], a [[Hopf algebra]], ''H'', is '''quasitriangular'''<ref>Montgomery & Schneider (2002), {{Google books |
In [[mathematics]], a [[Hopf algebra]], ''H'', is '''quasitriangular'''<ref>Montgomery & Schneider (2002), [{{Google books|plainurl=y|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>, |
Revision as of 18:57, 28 November 2014
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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 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.
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
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.