Skip to main content
SpringerLink
Log in
Book cover

Algebra, Arithmetic, and Geometry pp 549–566Cite as

Lie Algebra Theory without Algebra

  • Chapter
  • First Online:

Part of the book series: Progress in Mathematics ((PM,volume 269))

Summary

We present a self-contained and geometric proof of the standard results about maximal compact subgroups of simple Lie groups. The approach follows an idea suggested by Cartan, and is a variant of a proof given by Richardson in the complex case. We apply techniques associated to geometric invariant theory. The main step in the proof involves an argument with Riemannian metrics of nonpositive curvature.

2000 Mathematics Subject Classifications: 22 E15, 53C35

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   149.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD   199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. Bishop, R., O'Neill, B., Manifolds of negative curvature, Trans. Amer. Math. Soc. 145 (1969), 1–49.

    Article  MATH  MathSciNet  Google Scholar 

  2. Bogomolov, F.A., Holomorphic tensors and vector bundles on projective varities, Izv. Akad. nauk SSSR, Ser. Mat. 42 (1978), no. 6, 1227–1287 (English transl.: math. USSR, Izv, 13 (1979), 499–555).

    MathSciNet  Google Scholar 

  3. Bridson, M., Haefliger, A., Metric spaces of non-positive curvature, Springer (1999).

    Google Scholar 

  4. Cartan, E., Groupes simples clos et ouverts et géométrie Riemannienne, J. Math. Pures Appl. 8 (1929), 1–33.

    MathSciNet  Google Scholar 

  5. Helgason, S., Differential Geometry, Lie groups and symmetric spaces, Academic Press (1978).

    Google Scholar 

  6. Hesselink, W., Uniform instability in reductive groups, Jour. Reine Angew. Math. 304 (1978), 74–96.

    Article  MathSciNet  Google Scholar 

  7. Kirwan, F., Cohomology of quotients in symplectic and algebraic geometry, Princeton U.P. (1984).

    Google Scholar 

  8. Kempf, G., Instability in invariant theory, Annals of Math. 108 (1978), 299–316.

    Article  MathSciNet  Google Scholar 

  9. Kempf, G., Ness, L., The lengths of vectors in representation spaces, Springer Lecture Notes 732 (1978), 233–242.

    MathSciNet  Google Scholar 

  10. Mumford, D. Fogarty, J. Kirwan, F.C., Geometric Invariant Theory, 3rd. ed. Springer (1994).

    Google Scholar 

  11. Lauret, J., On the moment map for the variety of Lie algebras, Jour. Fuctional Analysis 202 (2003), 392–423.

    Article  MATH  MathSciNet  Google Scholar 

  12. Marian, A., On the real moment map, Math. Res. Letters 8 (2001), 779–788.

    MATH  MathSciNet  Google Scholar 

  13. Ness, L., A stratification of the null cone by the moment map, Amer. Jour. Math. 106 (1984), 128–132.

    Article  MathSciNet  Google Scholar 

  14. Richardson, R., Slodowy, P., Minimal vectors for real reductive group actions, Jour. Lond. Math. Soc. 42 (1990), 409–429.

    Article  MATH  Google Scholar 

  15. Richardson, R., Compact real forms of a complex semisimple Lie algebra, Jour. Differential Geometry 2 (1968), 411–420.

    MATH  Google Scholar 

  16. Rousseau, G., Immeubles, sphériques et théorie des invariants, C. R. Acad. Sci. Paris Sér. A 286 (1978), 247–250.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Imperial College, London, SW7 2AZ, UK

    Simon K. Donaldson

Authors
  1. Simon K. Donaldson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Simon K. Donaldson .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Courant Institute of Math. Sciences, New York University, Mercer St. 251, New York, 10012, New York, USA

    Yuri Tschinkel

  2. Eberly College of Science, Dept. Mathematics, Pennsylvania State University, University Park, 16802, Pennsylvania, USA

    Yuri Zarhin

Additional information

Dedicated to Professor Yu I. Manin, on his 70th birthday

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer Science+Business Media, LLC

About this chapter

Cite this chapter

Donaldson, S.K. (2009). Lie Algebra Theory without Algebra. In: Tschinkel, Y., Zarhin, Y. (eds) Algebra, Arithmetic, and Geometry. Progress in Mathematics, vol 269. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4745-2_12

Download citation

Publish with us

Policies and ethics

Search

Navigation