Skip to main content
Log in

Adams operations on the twisted K-theory of compact Lie groups

  • Published:
Journal of Homotopy and Related Structures Aims and scope Submit manuscript

Abstract

In this paper, extending the results in Fok (Proc Am Math Soc 145:2799–2813, 2017), we compute Adams operations on the twisted K-theory of connected, simply-connected and simple compact Lie groups G, in both equivariant and nonequivariant settings.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Atiyah, M.F.: Power operations in K-theory. Q. J. Math. 2(17), 165–93 (1966)

    Article  ADS  MathSciNet  Google Scholar 

  2. Atiyah, M.F., Hirzebruch, F.: Vector Bundles and Homogeneous Spaces. AMS Symposium in Pure Mathematics, III. American Mathematical Society, RI (1960)

    Google Scholar 

  3. Atiyah, M., Segal, G.: Twisted K-theory. Ukr. Mat. Visn. 1(3), 287–330 (2004)

    MathSciNet  Google Scholar 

  4. Atiyah, M.F., Segal, G.: Twisted K-Theory and Cohomology. Nankai Tracts in Mathematics: Inspired by S S Chern. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, pp. 5–43 (2006)

  5. Bousfield, A.K.: The K-theory localizations and \(\nu _1\)-periodic homotopy groups of H-spaces. Topology 38(6), 1239–1264 (1999)

    Article  MathSciNet  Google Scholar 

  6. Braun, V.: Twisted K-theory of Lie groups. J. High Energy Phys. 03, 029 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  7. Brylinski, J.-L., Zhang, B.: Equivariant K-theory of compact connected Lie groups. K-theory 20, 23–36 (2000)

    Article  MathSciNet  Google Scholar 

  8. Carey, A., Wang, B.-L.: Thom isomorphism and push-forward map in twisted K-theory. J. K-Theory K-Theory Appl. Algebra Geom. Topol. 1, 357–393 (2008)

    MathSciNet  Google Scholar 

  9. Douglas, C.: On the twisted K-homology of simple Lie groups. Topology 45, 955–988 (2006)

    Article  MathSciNet  Google Scholar 

  10. Fok, C.-K.: Adams operations on classical compact Lie groups. Proc. Am. Math. Soc. 145, 2799–2813 (2017)

    Article  MathSciNet  Google Scholar 

  11. Fok, C.-K.: Equivariant twisted real K-theory of compact Lie groups. J. Geom. Phys. 124, 325–349 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  12. Fok, C.-K.: Equivariant formality in K-theory. N. Y. J. Math. 25, 315–327 (2019)

    MathSciNet  Google Scholar 

  13. Freed, D.S., Hopkins, M.J., Teleman, C.: Loop groups and twisted K-theory III. Ann. Math. (2) 174(2), 947–1007 (2011)

    Article  MathSciNet  Google Scholar 

  14. Gepner, D.: Fusion rings and geometry. Commun. Math. Phys. 141, 381–411 (1991)

    Article  ADS  MathSciNet  Google Scholar 

  15. Hodgkin, L.: On the K-theory of Lie groups. Topology 6, 1–36 (1967)

    Article  MathSciNet  Google Scholar 

  16. Mathai, V., Rosenberg, J.: Group dualities, T-dualities, and twisted K-theory. J. Lond. Math. Soc. (2) 97(1), 1–23 (2018)

    Article  MathSciNet  Google Scholar 

  17. Meinrenken, E.: On the quantization of conjugacy classes. L’Enseignement Mathematique 55, 33–75 (2009)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We would like to thank the anonymous referees for the critical comments and especially the suggestions for improving the exposition and simplifying the proof of Corollary 3.6. We acknowledge support from the School of Mathematics and Physics Research Grant SRG2324-06 of the Xi’an Jiaotong-Liverpool University.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Chi-Kwong Fok.

Additional information

Communicated by Christopher Douglas.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fok, CK. Adams operations on the twisted K-theory of compact Lie groups. J. Homotopy Relat. Struct. (2024). https://doi.org/10.1007/s40062-024-00342-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40062-024-00342-9

Mathematics Subject Classification

Navigation