Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-10T17:02:57.799Z Has data issue: false hasContentIssue false
  • English
  • Français

Fibre product approach to index pairings for the generic Hopf fibration of SUq(2)

Published online by Cambridge University Press:  15 March 2010

Elmar Wagner
Elmar Wagner
Affiliation:
Instituto de Física y Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo, Morelia, México, Elmar.Wagner@math.uni-leipzig.de
Get access

Abstract

A fibre product construction is used to give a description of quantum line bundles over the generic Podlés spheres obtained by gluing two quantum discs along their boundaries. Representatives of the corresponding K0-classes are given in terms of 1-dimensional projections belonging to the C*-algebra, and in terms of analogues of the classical Bott projections. The K0-classes of quantum line bundles derived from the generic Hopf fibration of quantum SU(2) are determined and the index pairing is computed. It is argued that taking the projections obtained from the fibre product construction yields a significant simplification of earlier index computations.

Keywords

K0-groupquantum spheresquantum line bundlesindex pairing
Type
Research Article
Information
Journal of K-Theory , Volume 7 , Issue 1 , February 2011 , pp. 1 - 17
Copyright
Copyright © ISOPP 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Bass, H.: Algebraic K-theory, Benjamin, New York, 1968.Google Scholar
2.Baum, P. F., Hajac, P. M., Matthes, R., Szymański, W.: The K-theory of Heegaard-type quantum 3-spheres. K-Theory 35 (2005), 159–186.CrossRefGoogle Scholar
3.Brzeziński, T.: Quantum Homogeneous Spaces as Quantum Quotient Spaces. J. Math. Phys. 37 (1996), 2388–2399.CrossRefGoogle Scholar
4.Brzeziński, T., Hajac, P. M.: Coalgebra extensions and algebra coextensions of Galois type. Commun. Algebra 27 (1999), 1347–1367.CrossRefGoogle Scholar
5.Brzeziński, T., Majid, S.: Coalgebra bundles. Commun. Math. Phys. 191 (1998), 467–492.Google Scholar
6.Brzeziński, T., Majid, S.: Quantum geometry of algebra factorisations and coalgebra bundles. Commun. Math. Phys. 213 (2000), 491–521.CrossRefGoogle Scholar
7.Budzyński, R. J., Kondracki, W.: Quantum principal fibre bundles: Topological aspects. Rep. Math. Phys. 37 (1996), 365–385.CrossRefGoogle Scholar
8.Calow, D., Matthes, R.: Covering and gluing of algebras and differential algebras. J. Geom. Phys. 32 (2000), 364–396.CrossRefGoogle Scholar
9.Connes, A.: Noncommutative geometry. Academic Press, San Diego, 1994.Google Scholar
10.Dabrowski, L., Hadfield, T., Hajac, P. M., Matthes, R.: K-theoretic construction of noncommutative instantons of all charges. Preprint, arXiv: math/0702001v1.Google Scholar
11.Gracia-Bondía, J. M., Figueroa, H., Várilly, J. C.: Elements of Noncommutative Geometry. Birkhäuser, Boston, 2001.CrossRefGoogle Scholar
12.Hajac, P. M.: Bundles over quantum sphere and noncommutative index theorem. K-Theory 21 (1996), 141–150.CrossRefGoogle Scholar
13.Hajac, P. M., Majid, S.: Projective module description of the q-monopole. Commun. Math. Phys. 206 (1999), 247–264.CrossRefGoogle Scholar
14.Hajac, P. M., Matthes, R., Szymański, W.: Chern numbers for two families of noncommutative Hopf fibrations. C. R. Math. Acad. Sci. Paris 336 (2003), 925–930.CrossRefGoogle Scholar
15.Hajac, P. M., Matthes, R., Szymański, W.: Noncommutative index theory for mirror quantum spheres. C. R. Math. Acad. Sci. Paris 343 (2006), 731–736.CrossRefGoogle Scholar
16.Hajac, P. M., Wagner, E.: The pullbacks of principal coactions. In preparation.Google Scholar
17.Klimek, S., Lesniewski, A.: A two-parameter quantum deformation of the unit disc. J. Funct. Anal. 115 (1993), 1–23.CrossRefGoogle Scholar
18.Klimyk, K. A., Schmüdgen, K.: Quantum Groups and Their Representations. Springer, Berlin, 1997.CrossRefGoogle Scholar
19.Masuda, T., Nakagami, Y., Watanabe, J.: Noncommutative differential geometry on the quantum SU(2). I: An Algebraic Viewpoint. K-Theory 4 (1990), 157–180.CrossRefGoogle Scholar
20.Masuda, T., Nakagami, Y., Watanabe, J.: Noncommutative differential geometry on the quantum two sphere of Podlés. I: An Algebraic Viewpoint. K-Theory 5 (1991), 151–175.CrossRefGoogle Scholar
21.Müller, E. F., Schneider, H.-J.: Quantum homogeneous spaces with faithfully flat module structures. Israel J. Math. 111 (1999), 157–190.CrossRefGoogle Scholar
22.Podlés, P.: Quantum spheres. Lett. Math. Phys. 14 (1987), 193–202.CrossRefGoogle Scholar
23.Schmüdgen, K., Wagner, E.: Representations of cross product algebras of Podleś quantum spheres. J. Lie Theory 17 (2007), 751–790.Google Scholar
24.Sheu, A. J.-L.: Quantization of the Poisson SU(2) and its Poisson homogeneous space – the 2-sphere. Commun. Math. Phys. 135 (1991), 217–232.CrossRefGoogle Scholar