Letters in Mathematical Physics

, Volume 97, Issue 3, pp 263–277 | Cite as

Operator Algebra Quantum Homogeneous Spaces of Universal Gauge Groups

  • Snigdhayan MahantaEmail author
  • Varghese Mathai


In this paper, we quantize universal gauge groups such as SU(∞), as well as their homogeneous spaces, in the σ-C*-algebra setting. More precisely, we propose concise definitions of σ-C*-quantum groups and σ-C*-quantum homogeneous spaces and explain these concepts here. At the same time, we put these definitions in the mathematical context of countably compactly generated spaces as well as C*-compact quantum groups and homogeneous spaces. We also study the representable K-theory of these spaces and compute these groups for the quantum homogeneous spaces associated to the quantum version of the universal gauge group SU(∞).

Mathematics Subject Classification (2000)

46L80 58B32 58B34 46L65 


σ-C*-quantum groups σ-C*-quantum homogeneous spaces universal gauge groups operator algebras K-theory 


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  1. 1.
    Blackadar B.: Shape theory for C*-algebras. Math. Scand. 56(2), 249–275 (1985)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bourbaki N.: General Topology, Chaps. 1–4. Elements of Mathematics. Springer, New York (1998)Google Scholar
  3. 3.
    Bragiel K.: The twisted SU(N) group. On the C*-algebra C(S μ U(N)). Lett. Math. Phys. 20(3), 251–257 (1990)MathSciNetADSzbMATHCrossRefGoogle Scholar
  4. 4.
    Carey A.L., Mickelsson J.: The universal gerbe, Dixmier-Douady class, and gauge theory. Lett. Math. Phys. 59(1), 47–60 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Harvey J.A., Moore G.: Noncommutative tachyons and K-theory. Strings, branes, and M-theory. J. Math. Phys. 42(7), 2765–2780 (2001)MathSciNetADSzbMATHCrossRefGoogle Scholar
  6. 6.
    Hodgkin L.: On the K-theory of Lie groups. Topology 6, 1–35 (1967)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Kustermans J., Vaes S.: The operator algebra approach to quantum groups. Proc. Natl. Acad. Sci. USA 97(2), 547–552 (2000)MathSciNetADSzbMATHCrossRefGoogle Scholar
  8. 8.
    Kustermans, J., Vaes, S., Vainerman, L., Van Daele, A., Woronowicz, S.L.: Lecture Notes School/Conference on Noncommutative Geometry and Quantum Groups, Warsaw (2001).
  9. 9.
    Maldacena, J., Seiberg, N., Moore, G.: Geometrical interpretation of D-branes in gauged WZW models, J. High Energy Phys. 7, Paper 46, 63 pp (2001)Google Scholar
  10. 10.
    Mallios, A.: Topological algebras. Selected topics. In: North-Holland Mathematics Studies, vol. 124 (1984)Google Scholar
  11. 11.
    Nagy G.: On the Haar measure of the quantum SU(N) group. Commun. Math. Phys. 153, 217–228 (1993)ADSzbMATHCrossRefGoogle Scholar
  12. 12.
    Nagy G.: Deformation quantization and K-theory. Perspectives on quantization. Contemp. Math. 214, 111–134 (1998)CrossRefGoogle Scholar
  13. 13.
    Nagy G.: A deformation quantization procedure for C*-algebras. J. Oper. Theory 44(2), 369–411 (2000)zbMATHGoogle Scholar
  14. 14.
    Phillips N.C.: Inverse limits of C*-algebras. J. Oper. Theory 19(1), 159–195 (1988)zbMATHGoogle Scholar
  15. 15.
    Phillips, N.C.: Inverse limits of C*-algebras and applications. Operator algebras and applications, vol. 1, pp. 127–185. London. Math. Soc. Lecture Note Ser., vol. 135 (1988)Google Scholar
  16. 16.
    Phillips N.C.: Representable K-theory for σ-C*-algebras. K-Theory 3(5), 441–478 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Phillips N.C.: K-theory for Fréchet algebras. Int. J. Math. 2(1), 77–129 (1991)zbMATHCrossRefGoogle Scholar
  18. 18.
    Sheu A.: Compact quantum groups and groupoid C*-algebras. J. Funct. Anal. 144, 371–393 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Sheu A.: Quantum spheres as groupoid C*-algebras. Q. J. Math. Oxford Ser. (2) 48(192), 503–510 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Sheu A.: Groupoid approach to quantum projective spaces. Operator algebras and operator theory. Contemp. Math. 228(1), 341–350 (1998)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Soibelman Ya.S.: Algebra of functions on a compact quantum group and its representations. Algebra Analiz 2(1), 190–212 (1990)zbMATHGoogle Scholar
  22. 22.
    Soibelman Ya.S., Vaksman L.L.: The algebra of functions on the quantum group SU(n + 1), and odd-dimensional quantum spheres. Leningrad Math. J. 2, 1023–1042 (1991)MathSciNetGoogle Scholar
  23. 23.
    Weidner, J.: KK-groups for generalized operator algebras. I, II. K-Theory 3(1), 57–77, 79–98 (1989)Google Scholar
  24. 24.
    Woronowicz S.L.: A remark on compact matrix quantum groups. Lett. Math. Phys. 21(1), 35–39 (1991)MathSciNetADSzbMATHCrossRefGoogle Scholar
  25. 25.
    Woronowicz, S.L.: Compact quantum groups. Symétries quantiques (Les Houches), pp. 845–884. North-Holland, Amsterdam (1998)Google Scholar

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© Springer 2011

Authors and Affiliations

  1. 1.Department of Pure MathematicsUniversity of AdelaideAdelaideAustralia

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