Abstract
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(∞).
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
Blackadar B.: Shape theory for C*-algebras. Math. Scand. 56(2), 249–275 (1985)
Bourbaki N.: General Topology, Chaps. 1–4. Elements of Mathematics. Springer, New York (1998)
Bragiel K.: The twisted SU(N) group. On the C*-algebra C(S μ U(N)). Lett. Math. Phys. 20(3), 251–257 (1990)
Carey A.L., Mickelsson J.: The universal gerbe, Dixmier-Douady class, and gauge theory. Lett. Math. Phys. 59(1), 47–60 (2002)
Harvey J.A., Moore G.: Noncommutative tachyons and K-theory. Strings, branes, and M-theory. J. Math. Phys. 42(7), 2765–2780 (2001)
Hodgkin L.: On the K-theory of Lie groups. Topology 6, 1–35 (1967)
Kustermans J., Vaes S.: The operator algebra approach to quantum groups. Proc. Natl. Acad. Sci. USA 97(2), 547–552 (2000)
Kustermans, J., Vaes, S., Vainerman, L., Van Daele, A., Woronowicz, S.L.: Lecture Notes School/Conference on Noncommutative Geometry and Quantum Groups, Warsaw (2001). http://perswww.kuleuven.be/u0018768/artikels/lecture-notes.pdf
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)
Mallios, A.: Topological algebras. Selected topics. In: North-Holland Mathematics Studies, vol. 124 (1984)
Nagy G.: On the Haar measure of the quantum SU(N) group. Commun. Math. Phys. 153, 217–228 (1993)
Nagy G.: Deformation quantization and K-theory. Perspectives on quantization. Contemp. Math. 214, 111–134 (1998)
Nagy G.: A deformation quantization procedure for C*-algebras. J. Oper. Theory 44(2), 369–411 (2000)
Phillips N.C.: Inverse limits of C*-algebras. J. Oper. Theory 19(1), 159–195 (1988)
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)
Phillips N.C.: Representable K-theory for σ-C*-algebras. K-Theory 3(5), 441–478 (1989)
Phillips N.C.: K-theory for Fréchet algebras. Int. J. Math. 2(1), 77–129 (1991)
Sheu A.: Compact quantum groups and groupoid C*-algebras. J. Funct. Anal. 144, 371–393 (1997)
Sheu A.: Quantum spheres as groupoid C*-algebras. Q. J. Math. Oxford Ser. (2) 48(192), 503–510 (1997)
Sheu A.: Groupoid approach to quantum projective spaces. Operator algebras and operator theory. Contemp. Math. 228(1), 341–350 (1998)
Soibelman Ya.S.: Algebra of functions on a compact quantum group and its representations. Algebra Analiz 2(1), 190–212 (1990)
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)
Weidner, J.: KK-groups for generalized operator algebras. I, II. K-Theory 3(1), 57–77, 79–98 (1989)
Woronowicz S.L.: A remark on compact matrix quantum groups. Lett. Math. Phys. 21(1), 35–39 (1991)
Woronowicz, S.L.: Compact quantum groups. Symétries quantiques (Les Houches), pp. 845–884. North-Holland, Amsterdam (1998)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Mahanta, S., Mathai, V. Operator Algebra Quantum Homogeneous Spaces of Universal Gauge Groups. Lett Math Phys 97, 263–277 (2011). https://doi.org/10.1007/s11005-011-0492-y
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11005-011-0492-y