Quasitriangular Hopf algebra: Difference between revisions
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* {{cite book | last=Montgomery | first=Susan | authorlink=Susan Montgomery | title=Hopf algebras and their actions on rings | series=Regional Conference Series in Mathematics | volume=82 | location=Providence, RI | publisher=[[American Mathematical Society]] | year=1993 | isbn=0-8218-0738-2 | zbl=0793.16029 }} |
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* {{cite book |authorlink=Susan Montgomery |first=Susan | last=Montgomery | authorlink2=Hans-Jürgen Schneider |first2=Hans-Jürgen |last2=Schneider |title=New directions in Hopf algebras | series=Mathematical Sciences Research Institute Publications | volume=43 | publisher=[[Cambridge University Press]] | year=2002 | isbn=978-0-521-81512-3 | zbl=0990.00022 }} |
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Revision as of 18:35, 18 June 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 (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.