Talk:PlanetPhysics/Compact Quantum Groups

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\section{Compact Quantum Groups, (CQG) s}

A \emph{compact quantum group, $Q_{CG}$} is defined as a particular case of a \htmladdnormallink{locally compact quantum group}{http://planetphysics.us/encyclopedia/LCQG.html} $Q_{Glc}$ when the \htmladdnormallink{object}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} space of the latter $Q_{Glc}$ is a compact \htmladdnormallink{topological}{http://planetphysics.us/encyclopedia/CoIntersections.html} space (instead of being a locally compact one).

\textbf{Bibliography}

$[1]$ ABE, E., Hopf Algebras, Cambridge University Press, 1977.

$[2]$ BAAJ, S., SKANDALIS, G., Unitaires multiplicatifs et dualit\'e pour les produits crois\'es de C*-alg\'ebres, Ann. scient. Ec. \htmladdnormallink{Norm}{http://planetphysics.us/encyclopedia/NormInducedByInnerProduct.html}. Sup., 4e s\'erie, t. 26 (1993), 425-488.

$[3]$ CONWAY, J. B., A Course in Functional Analysis, Springer-Verlag, New York, 1985.

$[4]$ DIJKHUIZEN, M.S., KOORNWINDER, T.H., \htmladdnormallink{CQG}{http://planetphysics.us/encyclopedia/ClassicalTransformationGroup.html} algebras : a direct \htmladdnormallink{algebraic}{http://planetphysics.us/encyclopedia/CoIntersections.html} approach to \htmladdnormallink{quantum groups}{http://planetphysics.us/encyclopedia/QuantumGroup.html}, Lett. Math. Phys. 32 (1994), 315-330.\\ $[5]$ DIXMIER, J., \htmladdnormallink{C*-algebras}{http://planetphysics.us/encyclopedia/VonNeumannAlgebra2.html}, North-Holland Publishing Company, Amsterdam, 1982.\\ $[6]$ ENOCK, M., SCHWARTZ, J.-M., Kac Algebras and \htmladdnormallink{duality}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} of Locally Compact \htmladdnormallink{groups}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}, Springer-Verlag, Berlin (1992).\\ $[7]$ EFFROS, E.G., RUAN, Z.-J., Discrete Quantum Groups I. The \htmladdnormallink{Haar measure}{http://planetphysics.us/encyclopedia/HigherDimensionalQuantumAlgebroid.html}, Int. J. of Math. (1994), 681-723.\\ $[8]$ HOFMANN, K.H., Elements of compact semi-groups, Charles E. Merill Books Inc. Columbus, Ohio (1996).\\ $[9]$ HOLLEVOET, J., Lokaal compacte quantum-semigroepen : Representaties en Pontryagin-dualiteit, Ph.D. Thesis, K.U.Leuven, 1994.\\ $[10]$ HOLLEVOET, J., Pontryagin Duality for a Class of Locally Compact Quantum Groups, Math. Nachrichten 176 (1995), 93-110.\\ $[11]$ KIRCHBERG, E., Discrete Quantum Groups, talk at Oberwolfach, 1994.\\ $[12]$ KUSTERMANS, J., C*-algebraic Quantum Groups arising from Algebraic Quantum Groups, Ph.D. Thesis, K.U.Leuven, 1997.\\ $[13]$ KUSTERMANS, J., VAN DAELE, A., C*-algebraic Quantum Groups arising from Algebraic Quantum Groups, Int. J. of Math. 8 (1997), 1067-1139.\\ $[14]$ LANCE, E.C., An explicit description of the fundamental unitary for SU(2)q, Commun. Math. Phys. 164 (1994), 1-15.\\ $[15]$ DE MAGELHAES, I.V., Hopf-C*-algebras and locally compact groups, Pacific J. Math (2) 36 (1935), 448-463.\\ $[16]$ MASUDA, M., NAKAGAMI, Y., A \htmladdnormallink{von Neumann algebra}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra4.html} Framework for the Duality of Quantum Groups. Publications of the RIMS Kyoto University 30 (1994), 799-850.\\ $[17]$ MASUDA, M., A C*-algebraic framework for the quantum groups, talk at Warsaw workshop on Quantum Groups and Quantum Spaces, 1995.\\ $[18]$ MASUDA, M., NAKAGAMI, Y., WORONOWICZ,, S.L. (in preparation).\\ $[19]$ SHEU, A.J.L., Compact Quantum Groups and \htmladdnormallink{groupoid}{http://planetphysics.us/encyclopedia/LocallyCompactGroupoid.html} C*-Algebras, J. Funct. Analysis 144 (1997), 371-393.\\ $[20]$ SWEEDLER, M.E., Hopf Algebras, W.A. Benjamin, inc., New York, 1969.\\ $[21]$ TOMIYAMA, J., Applications of Fubini \htmladdnormallink{type}{http://planetphysics.us/encyclopedia/Bijective.html} \htmladdnormallink{theorems}{http://planetphysics.us/encyclopedia/Formula.html} to the \htmladdnormallink{tensor}{http://planetphysics.us/encyclopedia/Tensor.html} product of C*-algebras, Tokohu Math. J. 19 (1967), 213-226.\\ $[22]$ VAN DAELE, A., Dual Pairs of Hopf *-algebras, Bull. London Math. Soc. 25 (1993), 209-230.\\ $[23]$ VAN DAELE, A., Multiplier Hopf Algebras, Trans. Am. Math. Soc. 342 (1994), 917-932. $[24]$ VAN DAELE, A., The Haar Measure on a Compact Quantum Group, Proc. Amer. Math. Soc. 123 (1995), 3125-3128. $[25]$ VAN DAELE, A., Discrete Quantum Groups, Journal of Algebra 180 (1996), 431-444. $[26]$ VAN DAELE, A., An Algebraic Framework for Group Duality, preprint K.U.Leuven (1996), to appear in \emph{Advances of Mathematics.} \\ $[27]$ VAN DAELE, A., Multiplier Hopf Algebras and Duality, Proceedings of the workshop on Quantum Groups and Quantum Spaces in Warsaw (1995), Polish Academy of sciences Warszawa 40 (1997), 51-58.\\ $[28]$ VAN DAELE, A., The Haar measure on \htmladdnormallink{finite quantum groups}{http://planetphysics.us/encyclopedia/ComultiplicationInAQuantumGroup.html}, \emph{Proc. A.M.S.} 125 (1997), 3489-3500.\\ $[29]$ VAN DAELE, A., WANG, S., Universal Quantum Groups, \emph{Int. J. of Math.} (1996), 255-263. $[30]$ WANG, S., Krein Duality for Compact Quantum Groups, \emph{J. Math. Phys.} 38 (1997), 524-534.\\ 31. WORONOWICZ, S.L., Twisted $SU(2)$ group. An example of \htmladdnormallink{non-commutative}{http://planetphysics.us/encyclopedia/AbelianCategory3.html} differential calculus. Publ. RIMS Kyoto Univ. 23 No. 1 (1987), 117-181.\\ $[32]$ WORONOWICZ, S.L., Compact \htmladdnormallink{matrix}{http://planetphysics.us/encyclopedia/Matrix.html} Pseudogroups, \emph{Commun. Math. Phys.} 111 (1987), 613-665.\\ 33. WORONOWICZ, S.L., Tannaka-Krein duality for compact matrix pseudogroups. Twisted $SU(n)$ groups, Invent. Math. 93 (1988) 35-76. $[34]$ WORONOWICZ, S.L., A remark on Compact Matrix Quantum Groups, Lett. Math. Phys. 21 (1991), 35-39. $[35]$ WORONOWICZ, S.L., Compact Quantum Groups, Preprint University ofWarszawa (1992). To appear.\\ 36. MAES, A. and VanDAELE, A. 1998. \htmladdnormallink{Notes on Compact Quantum Groups.}{http://arxiv.org/PS_cache/math/pdf/9803/9803122v1.pdf}, $arxiv.org.math-FA-9803122v1$, 43 pp.

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