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|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) |
:*<math>R \ \Delta(x)R^{-1} = (T \circ \Delta)(x) </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>, |
:*<math>(\Delta \otimes 1)(R) = R_{13} \ R_{23}</math>, |
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:*<math>(1 \otimes \Delta)(R) = R_{13} \ R_{12}</math>, |
:*<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 |
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> |
:<math>\phi_{12}(a \otimes b) = a \otimes b \otimes 1,</math> |
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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]]. |
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As a consequence of the properties of quasitriangularity, the R-matrix, ''R'', is a solution of the [[ |
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<sup>2</sup>'' is an automorphism. In fact, ''S<sup>2</sup>'' 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 |
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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If the Hopf algebra ''H'' is quasitriangular, then the category of modules over ''H'' is braided with braiding |
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:<math>c_{U,V}(u\otimes v) = T \left( R \cdot (u \otimes v )\right) = T \left( R_1 u \otimes R_2 v\right) </math>. |
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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 |
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) \ |
:<math> (F \otimes 1) \cdot (\Delta \otimes id)( F) = (1 \otimes F) \cdot (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 |
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-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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* {{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]] |
[[Category:Hopf algebras]] |
Latest revision as of 18:29, 19 September 2023
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 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.
If the Hopf algebra H is quasitriangular, then the category of modules over H is braided with braiding
- .
Twisting
[edit]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
[edit]Notes
[edit]References
[edit]- 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.