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| A '''ribbon Hopf algebra''' <math>(A,m,\Delta,u,\varepsilon,S,\mathcal{R},\nu)</math> is a [[quasitriangular Hopf algebra]] which possess an invertible central element <math>\nu</math> more commonly known as the ribbon element, such that the following conditions hold:
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| :<math>\nu^{2}=uS(u), \; S(\nu)=\nu, \; \varepsilon (\nu)=1</math>
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| :<math>\Delta (\nu)=(\mathcal{R}_{21}\mathcal{R}_{12})^{-1}(\nu \otimes \nu )</math>
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| where <math>u=m(S\otimes \text{id})(\mathcal{R}_{21})</math>. Note that the element ''u'' exists for any quasitriangular Hopf algebra, and
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| <math>uS(u)</math> must always be central and satisfies <math>S(uS(u))=uS(u), \varepsilon(uS(u))=1, \Delta(uS(u)) =
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| (\mathcal{R}_{21}\mathcal{R}_{12})^{-2}(uS(u) \otimes uS(u))</math>, so that all that is required is that it have a central square root with the above properties.
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| Here
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| :<math> A </math> is a vector space
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| :<math> m </math> is the multiplication map <math>m:A \otimes A \rightarrow A</math>
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| :<math> \Delta </math> is the co-product map <math>\Delta: A \rightarrow A \otimes A</math>
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| :<math> u </math> is the unit operator <math>u:\mathbb{C} \rightarrow A</math>
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| :<math> \varepsilon </math> is the co-unit operator <math>\varepsilon: A \rightarrow \mathbb{C}</math>
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| :<math> S </math> is the antipode <math>S: A\rightarrow A</math>
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| :<math>\mathcal{R}</math> is a universal R matrix
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| We assume that the underlying field <math>K</math> is <math>\mathbb{C}</math>
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| == See also ==
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| *[[Quasitriangular Hopf algebra]]
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| *[[Quasi-triangular Quasi-Hopf algebra]]
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| == References ==
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| *{{cite journal |last=Altschuler |first=D. |last2=Coste |first2=A. |title=Quasi-quantum groups, knots, three-manifolds and topological field theory |journal=[[Communications in Mathematical Physics|Commun. Math. Phys.]] |volume=150 |year=1992 |issue= |pages=83–107 |id={{ArXiv|hep-th|9202047}} }}
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| *{{cite book |last=Chari |first=V. C. |last2=Pressley |first2=A. |title=A Guide to Quantum Groups |publisher=Cambridge University Press |year=1994 |isbn=0-521-55884-0 }}
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| *{{cite journal |authorlink=Vladimir Drinfeld |first=Vladimir |last=Drinfeld |title=Quasi-Hopf algebras |journal=Leningrad Math J. |volume=1 |year=1989 |pages=1419–1457 }}
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| *{{cite book |first=Shahn |last=Majid |title=Foundations of Quantum Group Theory |publisher=Cambridge University Press |year=1995 }}
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| [[Category:Hopf algebras]]
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