Quasitriangular Hopf algebra: Difference between revisions
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:<math>\phi_{23}(a \otimes b) = 1 \otimes a \otimes b.</math> |
:<math>\phi_{23}(a \otimes b) = 1 \otimes a \otimes b.</math> |
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''R'' is called the R-matrix |
''R'' is called the R-matrix or 8==D |
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As a consequence of the properties of quasitriangularity, the R-matrix, ''R'', is a solution of the [[Yang-Baxter equation]] (and so a [[Module (mathematics)|module]] ''V'' of ''H'' can be used to determine quasi-invariants of [[braid theory|braids]], [[knot (mathematics)|knots]] and [[link (knot theory)|links]]). Also as a consequence of the properties of quasitriangularity, <math>(\epsilon \otimes 1) R = (1 \otimes \epsilon) R = 1 \in H</math>; moreover |
As a consequence of the properties of quasitriangularity, the R-matrix, ''R'', is a solution of the [[Yang-Baxter equation]] (and so a [[Module (mathematics)|module]] ''V'' of ''H'' can be used to determine quasi-invariants of [[braid theory|braids]], [[knot (mathematics)|knots]] and [[link (knot theory)|links]]). Also as a consequence of the properties of quasitriangularity, <math>(\epsilon \otimes 1) R = (1 \otimes \epsilon) R = 1 \in H</math>; moreover |
Revision as of 10:35, 18 September 2008
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 morphisms determined by
R is called the R-matrix or 8==D
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.