Ribbon Hopf algebra: Difference between revisions

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A ribbon Hopf algebra $(A,\nabla ,\eta ,\Delta ,\varepsilon ,S,{\mathcal {R}},\nu )$ is a quasitriangular Hopf algebra which possess an invertible central element $\nu$ more commonly known as the ribbon element, such that the following conditions hold:

\nu ^{2}=uS(u),\;S(\nu )=\nu ,\;\varepsilon (\nu )=1

\Delta (\nu )=({\mathcal {R}}_{21}{\mathcal {R}}_{12})^{-1}(\nu \otimes \nu )

where $u=\nabla (S\otimes {\text{id}})({\mathcal {R}}_{21})$ . Note that the element u exists for any quasitriangular Hopf algebra, and $uS(u)$ must always be central and satisfies $S(uS(u))=uS(u),\varepsilon (uS(u))=1,\Delta (uS(u))=({\mathcal {R}}_{21}{\mathcal {R}}_{12})^{-2}(uS(u)\otimes uS(u))$ , so that all that is required is that it have a central square root with the above properties.

Here

A

is a vector space

\nabla

is the multiplication map

\nabla :A\otimes A\rightarrow A

\Delta

is the co-product map

\Delta :A\rightarrow A\otimes A

\eta

is the unit operator

\eta :\mathbb {C} \rightarrow A

\varepsilon

is the co-unit operator

\varepsilon :A\rightarrow \mathbb {C}

S

is the antipode

S:A\rightarrow A

{\mathcal {R}}

is a universal R matrix

We assume that the underlying field $K$ is $\mathbb {C}$

If $A$ is finite-dimensional, one could equivalently call it ribbon Hopf if and only if its category of (say, left) modules is ribbon; if $A$ is finite-dimensional and quasi-triangular, then it is ribbon if and only if its category of (say, left) modules is pivotal.

References

Altschuler, D.; Coste, A. (1992). "Quasi-quantum groups, knots, three-manifolds and topological field theory". Commun. Math. Phys. 150 (1): 83–107. arXiv:hep-th/9202047. Bibcode:1992CMaPh.150...83A. doi:10.1007/bf02096567.
Chari, V. C.; Pressley, A. (1994). A Guide to Quantum Groups. Cambridge University Press. ISBN 0-521-55884-0.
Drinfeld, Vladimir (1989). "Quasi-Hopf algebras". Leningrad Math J. 1: 1419–1457.
Majid, Shahn (1995). Foundations of Quantum Group Theory. Cambridge University Press.

@@ Line 1: / Line 1: @@
+{{Short description|Algebraic structure}}
-A '''Ribbon Hopf algebra''' <math>(A,m,\Delta,u,\varepsilon,\mathcal{R},\nu)</math> is a [[Quasi-triangular Hopf algebra]]
-which possess an invertible central element <math>\nu</math> more commonly known as the ribbon element, such that the following conditions hold:
+A '''ribbon Hopf algebra''' <math>(A,\nabla, \eta,\Delta,\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:
 :<math>\nu^{2}=uS(u), \; S(\nu)=\nu, \; \varepsilon (\nu)=1</math>
 :<math>\Delta (\nu)=(\mathcal{R}_{21}\mathcal{R}_{12})^{-1}(\nu \otimes \nu )</math>
-such that <math>u=m(S\otimes id)(\mathcal{R}_{21})</math>.
+where <math>u=\nabla(S\otimes \text{id})(\mathcal{R}_{21})</math>.  Note that the element ''u'' exists for any quasitriangular Hopf algebra, and
+<math>uS(u)</math> must always be central and satisfies <math>S(uS(u))=uS(u), \varepsilon(uS(u))=1, \Delta(uS(u)) =
+(\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.
+Here
-Where
 :<math> A </math> is a vector space
-:<math> m </math> is the multiplication map <math>m:A \otimes A \rightarrow A</math>
+:<math> \nabla </math> is the multiplication map <math>\nabla:A \otimes A \rightarrow A</math>
 :<math> \Delta </math> is the co-product map <math>\Delta: A \rightarrow A \otimes A</math>
-:<math> u </math> is the unit operator <math>u:\mathbb{C} \rightarrow A</math>
+:<math> \eta </math> is the unit operator <math>\eta:\mathbb{C} \rightarrow A</math>
-:<math> \varepsilon </math> is the co-unit opertor <math>\varepsilon: A \rightarrow \mathbb{C}</math>
+:<math> \varepsilon </math> is the co-unit operator <math>\varepsilon: A \rightarrow \mathbb{C}</math>
+:<math> S </math> is the antipode <math>S: A\rightarrow A</math>
 :<math>\mathcal{R}</math> is a universal R matrix
 We assume that the underlying field <math>K</math> is <math>\mathbb{C}</math>
+If <math> A </math> is finite-dimensional, one could equivalently call it ''ribbon Hopf'' if and only if its category of (say, left) modules is ribbon; if <math> A </math> is finite-dimensional and quasi-triangular, then it is ribbon if and only if its category of (say, left) modules is pivotal.
 == See also ==
 *[[Quasitriangular Hopf algebra]]
-*[[Quasi-triangular Quasi-Hopf algebra]]
+*[[Quasi-triangular quasi-Hopf algebra]]
 == References ==
-* Altschuler, D., Coste, A.: Quasi-quantum groups, knots, three-manifolds and topological field theory. Commun. Math. Phys. 150 1992 83-107 http://arxiv.org/pdf/hep-th/9202047
+*{{cite journal |last1=Altschuler |first1=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=1 |pages=83–107 |arxiv=hep-th/9202047 |doi=10.1007/bf02096567|bibcode=1992CMaPh.150...83A }}
-* Chari, V.C., Pressley, A.: ''A Guide to Quantum Groups'' Cambridge University Press, 1994 ISBN 0-521-55884-0.
+*{{cite book |last1=Chari |first1=V. C. |last2=Pressley |first2=A. |title=A Guide to Quantum Groups |url=https://archive.org/details/guidetoquantumgr0000char |url-access=registration |publisher=Cambridge University Press |year=1994 |isbn=0-521-55884-0 }}
-* [[Vladimir Drinfeld]], ''Quasi-Hopf algebras'', Leningrad Math J. 1 (1989), 1419-1457
+*{{cite journal |author-link=Vladimir Drinfeld |first=Vladimir |last=Drinfeld |title=Quasi-Hopf algebras |journal=Leningrad Math J. |volume=1 |year=1989 |pages=1419–1457 }}
-* Majid, S.: ''Foundations of Quantum Group Theory'' Cambridge University Press, 1995
+*{{cite book |first=Shahn |last=Majid |title=Foundations of Quantum Group Theory |publisher=Cambridge University Press |year=1995 }}
 [[Category:Hopf algebras]]