On a Certain Approach to Quantum Homogeneous Spaces

Springer Science and Business Media LLC - Tập 313 - Trang 237-255 - 2012
P. Kasprzak1
1Department of Mathematical Methods in Physics, Faculty of Physics, Warsaw University, Warsaw, Poland

Tóm tắt

We propose a definition of a quantum homogeneous space of a locally compact quantum group. We show that classically it reduces to the notion of homogeneous spaces, giving rise to an operator algebraic characterization of the transitive group actions. On the quantum level our definition goes beyond the quotient case providing a framework which, besides the Vaes’ quotient of a locally compact quantum group by its closed quantum subgroup (our main motivation) is also compatible with, generically non-quotient, quantum homogeneous spaces of a compact quantum group studied by P. Podleś as well as the Rieffel deformation of G-homogeneous spaces. Finally, our definition rules out the paradoxical examples of the non-compact quantum homogeneous spaces of a compact quantum group.

Tài liệu tham khảo

Baaj S., Skandalis G.: Unitaires multiplicatifs et dualité pour les produits croisés de C*-algebres. Ann. Sci. Ec. Nor. Sup. 26(4), 425–488 (1993)

Bichon J., Rijdt A.D., Vaes S.: Ergodic coactions with large multiplicity and monoidal equivalence of quantum groups. Commun. Math. Phys. 262, 703–728 (2006)

Boca F.P.: Ergodic actions of compact matrix pseudogroups on C*-algebras. Rec. Adv. Op. Alg., Asterisque 232, 93–110 (1995)

Buss A., Meyer R.: Square-integrable coactions of locally compact quantum groups. Rep. Math. Phys. 63(2), 191–224 (2009)

de Commer K.: Galois objects and cocycle twisting for locally compact quantum groups. J. Op. Theory. 66(1), 59–106 (2011)

Enock M., Vainerman L.: Deformation of a Kac algebra by an abelian subgroup. Commun. Math. Phys. 178(3), 571–596 (1996)

Fima P., Vainerman L.: Twisting and Rieffel’s deformation of locally compact quantum groups. Deformation of the Haar measure. Commun. Math. Phys. 286(3), 1011–1050 (2009)

Kaliszewski S., Quigg J.: Categorical Landstad duality for actions. Indiana Univ. Math. J. 58(1), 415–441 (2009)

Kasprzak P.: Rieffel deformation of group coactions. Commun. Math. Phys. 300(3), 741–763 (2010)

Kasprzak P.: Rieffel deformation of homogeneous spaces. J. Funct. Anal. 260(1), 146–163 (2011)

Kasprzak P.: Rieffel deformation via crossed products. J. Funct. Anal. 257(5), 1288–1332 (2009)

Kasprzak, P., Sołtan, P.M.: Embeddable quantum homogeneous spaces. In preparation

Kleppner A.: Multipliers on Abelian Groups. Math. Ann. 158, 11–34 (1965)

Kustermans J., Vaes S.: Locally compact quantum groups. Ann. Sci. Ec. Norm. Sup. 33(4), 837–934 (2000)

Kustermans J., Vaes S.: Locally compact quantum groups in the von Neumann algebraic setting. Math. Scandin. 92(1), 68–92 (2003)

Lance, E.C.: Hilbert C*-modules. A toolkit for operator algebraists. Lon. Math. Soc. Lec. Notes 210, Cambridge: Cambridge University Press, 1995

Landstad M.B.: Quantizations arising from abelian subgroups. Internat. J. Math. 5(6), 897–936 (1994)

Masuda T., Nakagami Y., Woronowicz S.L.: A C*-algebraic framework for quantum groups. Internat. J. Math. 14(9), 903–1001 (2003)

Podleś P.: Quantum spheres. Lett. Math. Phys. 14(3), 193–202 (1987)

Podleś P.: Symmetries of quantum spaces. Subgroups and quotient spaces of quantum SU(2) and SO(3) groups. Subgroups. Commun. Math. Phys. 170(1), 1–20 (1995)

Rieffel, M.A.: Deformation quantization for action of \({\mathbb{R}^d}\). Mem. Am. Math. Soc. 106(506) (1993)

Rieffel M.A.: Non-compact quantum groups associated with abelian subgroups. Commun. Math. Phys. 171(1), 181–201 (1995)

Rieffel, M.A.: Proper actions of groups on C*-algebras. In: Mappings of operator algebras (Philadelphia 1988), Progr. Math. 84, Boston, MA: Birkhauser, 1990, pp. 141–182

Sołtan P.M.: Examples of non-compact quantum group actions. J. Math. Anal. Appl. 372, 224–236 (2010)

Sunder, V.S.: An invitation to von Neumann algebras. Universitext, Berlin-Heidelberg-New York: Springer-Verlag, 1986

Vaes S.: A new approach to induction and imprimitivity results. J. Funct. An. 229(2), 317–374 (2005)

Vaes S.: The unitary implementation of a locally compact quantum group action. J. Funct. An. 180, 426–480 (2001)

Wang S.: Ergodic actions of universal quantum groups on operator algebras. Commun. Math. Phys. 203, 481–498 (1999)

Woronowicz S.L.: C*-algebras generated by unbounded elements. Rev. Math. Phys. 7(3), 481–521 (1995)

Woronowicz, S.L.: Compact Quantum Groups. In: Symmétries quantiques (Les Houches 1995), Amsterdam: North-Holland, 1998, pp. 845–884

Woronowicz S.L.: From multiplicative unitaries to quantum groups. Int. J. Math. 7(1), 127–149 (1996)

Woronowicz S.L.: Quantum SU(2) and E(2) groups. Contraction procedure. Commun. Math. Phys. 149, 637–652 (1992)

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237-255

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