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| 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>,
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| :*<math>(\Delta \otimes 1)(R) = R_{13} \ R_{23}</math>,
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| :*<math>(1 \otimes \Delta)(R) = R_{13} \ R_{12}</math>,
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| where <math>R_{12} = \phi_{12}(R)</math>, <math>R_{13} = \phi_{13}(R)</math>, and <math>R_{23} = \phi_{23}(R)</math>, where <math>\phi_{12} : H \otimes H \to H \otimes H \otimes H</math>, <math>\phi_{13} : H \otimes H \to H \otimes H \otimes H</math>, and <math>\phi_{23} : H \otimes H \to H \otimes H \otimes H</math>, are algebra [[morphism]]s determined by
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| :<math>\phi_{12}(a \otimes b) = a \otimes b \otimes 1,</math>
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| :<math>\phi_{13}(a \otimes b) = a \otimes 1 \otimes b,</math>
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| :<math>\phi_{23}(a \otimes b) = 1 \otimes a \otimes b.</math>
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| ''R'' is called the R-matrix.
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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
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| <math>R^{-1} = (S \otimes 1)(R)</math>, <math>R = (1 \otimes S)(R^{-1})</math>, and <math>(S \otimes S)(R) = R</math>. One may further show that the
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| 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: <math>S^2(x)= 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]] quantum double construction.
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| ==Twisting==
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| The property of being a [[quasi-triangular Hopf algebra]] is preserved by [[Quasi-bialgebra#Twisting|twisting]] via an invertible element <math> F = \sum_i f^i \otimes f_i \in \mathcal{A \otimes A} </math> such that <math> (\varepsilon \otimes id )F = (id \otimes \varepsilon)F = 1 </math> and satisfying the cocycle condition
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| :<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 Drinfeld) twist.
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| ==See also==
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| * [[Quasi-triangular Quasi-Hopf algebra]]
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| * [[Ribbon Hopf algebra]]
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| == Notes ==
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| <references/>
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| == References ==
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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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| {{DEFAULTSORT:Quasitriangular Hopf Algebra}}
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| [[Category:Hopf algebras]]
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The writer is known by the name of Numbers Lint. For a whilst she's been in South Dakota. Bookkeeping is her working day job now. To gather coins is one of the issues I adore most.
Here is my web-site; at home std test