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%%% Primary Title: Hopf algebras and quantum groups
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\begin{document}
\subsection{Quantum group definitions}
\subsubsection{Background}
A quantum `group' is often defined as a {\em \htmladdnormallink{Hopf algebra}{http://planetphysics.us/encyclopedia/QuantumGroup.html}} or coalgebra. Actually, the duals of commutative Hopf algebras obtained by Fourier transformation are finite \htmladdnormallink{compact quantum groups}{http://planetphysics.us/encyclopedia/CompactQuantumGroups.html} that are Abelian.
Let us consider next, alternative definitions of quantum groups that indeed possess \htmladdnormallink{extended quantum symmetries}{http://planetphysics.us/encyclopedia/ExtendedQuantumSymmetries.html} and \htmladdnormallink{algebraic}{http://planetphysics.us/encyclopedia/CoIntersections.html} properties distinct from those of Hopf algebras.
\textbf{Quantum \htmladdnormallink{groups}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}, \htmladdnormallink{quantum operator algebras}{http://planetphysics.us/encyclopedia/Groupoid.html} and Related Symmetries.}
\begin{definition} {\em quantum groups} are defined as locally compact \htmladdnormallink{topological groups}{http://planetphysics.us/encyclopedia/PolishGroup.html} endowed with
a left \htmladdnormallink{Haar measure}{http://planetphysics.us/encyclopedia/HigherDimensionalQuantumAlgebroid.html} \htmladdnormallink{system}{http://planetphysics.us/encyclopedia/GenericityInOpenSystems.html}, and also with at least one internal \htmladdnormallink{quantum symmetry}{http://planetphysics.us/encyclopedia/HilbertBundle.html}, such as the intrinsic \htmladdnormallink{spin}{http://planetphysics.us/encyclopedia/QuarkAntiquarkPair.html} symmetry represented by either Pauli \htmladdnormallink{matrices}{http://planetphysics.us/encyclopedia/Matrix.html} or the Dirac algebra of \htmladdnormallink{observable}{http://planetphysics.us/encyclopedia/QuantumSpinNetworkFunctor2.html} spin \htmladdnormallink{operators}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra4.html}.
\end{definition}
For additional examples of quantum groups the reader is referred to the last six publications listed in the bibliography.
\textbf{Remark:}
One can also consider quantum groups as a particular case of {\em \htmladdnormallink{quantum groupoids}{http://planetphysics.us/encyclopedia/QuantumGroupoids.html}} in the limiting
case where there is only one symmetry \htmladdnormallink{type}{http://planetphysics.us/encyclopedia/Bijective.html} present in the quantum groupoid.
\subsubsection{Background to Quantum Groups, Paragroups and Operator Algebras in Quantum Theories}
\htmladdnormallink{Quantum theories}{http://planetphysics.us/encyclopedia/SpaceTimeQuantizationInQuantumGravityTheories.html} adopted a new lease of life post 1955 when von Neumann beautifully re-formulated \htmladdnormallink{quantum mechanics}{http://planetphysics.us/encyclopedia/QuantumParadox.html} (\htmladdnormallink{QM}{http://planetphysics.us/encyclopedia/FTNIR.html}) in the mathematically rigorous context of \htmladdnormallink{Hilbert spaces}{http://planetphysics.us/encyclopedia/NormInducedByInnerProduct.html} and operator
algebras. From a current physics perspective, von Neumann's
approach to quantum mechanics has done however much more: it has
not only paved the way to expanding the role of symmetry in
physics, as for example with the Wigner-Eckhart \htmladdnormallink{theorem}{http://planetphysics.us/encyclopedia/Formula.html} and its
applications, but also revealed the fundamental importance in
quantum physics of the \htmladdnormallink{state space}{http://planetphysics.us/encyclopedia/StableAutomaton.html} geometry of (quantum) operator
algebras. Subsequent developments of the \htmladdnormallink{quantum operator algebra}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra4.html} were aimed at identifying more general quantum symmetries than those defined for example by \htmladdnormallink{symmetry groups}{http://planetphysics.us/encyclopedia/TopologicalOrder2.html}, groups of unitary operators and \htmladdnormallink{Lie groups}{http://planetphysics.us/encyclopedia/BilinearMap.html}. Several fruitful quantum algebraic \htmladdnormallink{concepts}{http://planetphysics.us/encyclopedia/PreciseIdea.html} were developed, such as: the Ocneanu \textit{paragroups}-later found to be represented by Kac--Moody algebras, quantum `groups' represented either
as Hopf algebras or locally compact groups with Haar measure, `quantum' groupoids represented as \htmladdnormallink{weak Hopf algebras}{http://planetphysics.us/encyclopedia/WeakHopfAlgebra.html}, and so on. The Ocneanu {\em \htmladdnormallink{paragroups}{http://planetphysics.us/encyclopedia/Paragroups.html}} case is particularly interesting as it can be considered as an extension through \htmladdnormallink{quantization}{http://planetphysics.us/encyclopedia/MoyalDeformation.html} of certain finite group symmetries to infinitely-dimensional von Neumann type $II_1$ factors (subalgebras), and are, in effect, \textit{`quantized groups'} that can be nicely constructed as Kac algebras; in fact, it was recently shown that a paragroup can be constructed from a crossed product by an outer action of a Kac algebra. This suggests a \htmladdnormallink{relation}{http://planetphysics.us/encyclopedia/Bijective.html} to categorical aspects of paragroups (rigid monoidal \htmladdnormallink{tensor}{http://planetphysics.us/encyclopedia/Tensor.html} \htmladdnormallink{categories}{http://planetphysics.us/encyclopedia/Cod.html} previously reported in the literature). The strict symmetry of the group of (quantum) unitary operators is thus naturally extended through paragroups to the symmetry of the latter structure's unitary \htmladdnormallink{representations}{http://planetphysics.us/encyclopedia/CategoricalGroupRepresentation.html}; furthermore, if a subfactor of the \htmladdnormallink{von Neumann algebra}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra4.html} arises as a crossed product by a finite group action, the paragroup for this subfactor contains a very similar group structure to that of the original finite group, and also has a unitary representation theory similar to that of the original finite group. Last-but-not least, a paragroup yields a \emph{complete invariant} for irreducible inclusions of AFD von Neumannn type $II_1$ factors with finite index and finite depth (Theorem 2.6. of Sato, 2001). This can be considered as a kind of internal, `hidden' quantum symmetry of von Neumann algebras.
On the other hand, unlike paragroups, (quantum) locally compact groups are not readily constructed as either Kac or Hopf \htmladdnormallink{C*-algebras}{http://planetphysics.us/encyclopedia/VonNeumannAlgebra2.html}. In recent years the techniques of Hopf symmetry and
those of weak Hopf C*-algebras, sometimes called \emph{quantum `groupoids'} (cf B\"ohm et al.,1999),
provide important tools--in addition to the paragroups-- for studying the broader relationships of the
Wigner fusion rules algebra, $6j$--symmetry (Rehren, 1997), as well as the study of the \htmladdnormallink{noncommutative}{http://planetphysics.us/encyclopedia/AbelianCategory3.html} symmetries of subfactors within the Jones tower constructed from finite index depth 2 inclusion of factors,
also recently considered from the viewpoint of related Galois correspondences (Nikshych and Vainerman, 2000).
\begin{thebibliography}{9}
\bibitem{Chaician}
M. Chaician and A. Demichev: \emph{Introduction to Quantum Groups}, World Scientific (1996).
\bibitem{Drinfeld}
V. G. Drinfel'd: Quantum groups, In \emph{Proc. Intl. Congress of
Mathematicians, Berkeley 1986}, (ed. A. Gleason), Berkeley, 798-820 (1987).
\bibitem{Etingof1}
P.. I. Etingof and A. N. Varchenko, Solutions of the Quantum Dynamical Yang-Baxter Equation and Dynamical Quantum Groups, \emph{Comm.Math.Phys.}, \textbf{196}: 591-640 (1998).
\bibitem{Etingof2}
P. I. Etingof and A. N. Varchenko: Exchange dynamical quantum groups, \emph{Commun. Math. Phys.} \textbf{205} (1): 19-52 (1999)
\bibitem{Etingof3}
P. I. Etingof and O. Schiffmann: Lectures on the dynamical Yang--Baxter equations, in \emph{Quantum Groups and Lie Theory (Durham, 1999)}, pp. 89-129, Cambridge University Press, Cambridge, 2001.
\bibitem{Fell}
J. M. G. Fell.: The Dual Spaces of C*--Algebras., \emph{Transactions of the American
Mathematical Society}, \textbf{94}: 365--403 (1960).
\bibitem{Hahn1}
P. Hahn: Haar measure for measure groupoids., \textit{Trans. Amer. Math. Soc}. \textbf{242}: 1--33(1978).
\bibitem{Hahn2}
P. Hahn: The regular representations of measure groupoids., \textit{Trans. Amer. Math. Soc}. \textbf{242}:34--72(1978).
\bibitem{HLS2k8}
C. Heunen, N. P. Landsman, B. Spitters.: A topos for algebraic quantum theory, (2008) \\
arXiv:0709.4364v2 [quant--ph]
\bibitem{Majid}
S. Majid. Quantum groups, \htmladdnormallink{on line}{http://www.ams.org/notices/200601/what-is.pdf}
\end{thebibliography}
\end{document}
