Skip to main content
SpringerLink
Log in

A Characterization of the Classical Groups

  • Chapter
  • First Online:
Collected Papers

  • 2339 Accesses

Abstract

By one method of classification there are three types of (complex, connected) classical groups, (a) GL(n. C), (b) SO(n, C), and (c) Sp(n, C). So designated, each type is given as a specific group of matrices. It is perhaps neater (and for us more pertinent) to describe these groups by means of the special linear representation which each type admits.

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

Access this chapter

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 189.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Cloth bound cover Book
USD 249.99
Price excludes VAT (USA)
  • 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

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. S. Bochner and K. Yano, curvature and betti numbers, Annals of Mathematics Studies, no. 32, Princeton, 1953.

    Google Scholar 

  2. A. Borel and I. de Siebenthal, les sous-groupes feme’s de rang maximum des groupes de lie clos, Commentarii Mathematici Helvetici, vol. 23(1949), pp. 200–221.

    Article  MathSciNet  Google Scholar 

  3. E. Cartan, legons sur la geometrie des espaces de riemann, Paris, 1946.

    Google Scholar 

  4. E. Cartan, sur une classe remarquable d’espaces de riemann. I, Bulletin de la Societe Mathematique de France, vol. 54(1926), pp. 214–264.

    Article  MathSciNet  Google Scholar 

  5. E. Cartan, sur une classe remarquable d’espaces de riemann, I, II, Bulletin de la Societe Mathematique de France, vol. 55(1927), pp. 114–134.

    MathSciNet  Google Scholar 

  6. E. Cartan, la. g6ome’trie des groupes de transformations, Journal de Mathematiques Pures et Appliquees, vol. 6(1927), pp. 1–119.

    Google Scholar 

  7. E. B. Dynkin, the structure of semi-simple algebras, Uspehi Matematicheskih Nauk (N.S.)2, no. 4(20), (1947), pp. 59–127. American Mathematical Society Translation no. 17.

    MathSciNet  Google Scholar 

  8. H. Freudenthal, zur berechnung der charaktere der halbeinfachen lieschen gruppen. I, II, Koninklijke Nederlandse Akademie van Wetenschappen. Proceedings of the Section of Sciences, no. 4, vol. 57(1954), pp. 369–376.

    MathSciNet  Google Scholar 

  9. F. Gantmacheb, Canonical representation of automorphisms of a complex semi-simple Lie group, Matematiceski Sbornik, vol. 47(1939), pp. 101–146.

    MathSciNet  Google Scholar 

  10. B. Kostant, Holonomy and the Lie algebra of infinitesimal motions of a Riemannian mani-fold, Transactions of the American Mathematical Society, vol. 80 (1955), pp. 528–542.

    Article  MathSciNet  Google Scholar 

  11. J-L. Kosztjl, Homologie et cohomologie des alghbres de Lie, Bulletin de la Societe Mathe-matique de France, vol. 78(1950), pp. 65–127.

    Google Scholar 

  12. A. I. Malcev, On semi-simple subgroups of Lie groups, Bulletin of the Academy of Sciences URSS, Series on Mathematics, vol. 8(1944), pp. 143–174. American Mathematical Society Translation no. 33.

    MathSciNet  Google Scholar 

  13. Seminaire Sophus Lie, le annee 1954/1955, Thforie des algbbre de Lie, Topologie des groupes de Lie, Ecole Normale Superieure, Paris, 1955.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

    Bertram Kostant

Authors
  1. Bertram Kostant
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bertram Kostant .

Editor information

Editors and Affiliations

  1. Fac. Mathematics & Computer Science, Weizmann Institute of Science, Rehovot, 76100, Israel

    Anthony Joseph

  2. Dept. Mathematics, University of North Carolina, Phillips Hall, Chapel Hill, 27599-3250, USA

    Shrawan Kumar

  3. Centre de Mathématiques, Ecole Polytechnique, Palaiseau Cedex, 91128, France

    Michèle Vergne

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag New York

About this chapter

Cite this chapter

Kostant, B. (2009). A Characterization of the Classical Groups. In: Joseph, A., Kumar, S., Vergne, M. (eds) Collected Papers. Springer, New York, NY. https://doi.org/10.1007/b94535_9

Download citation

Publish with us

Policies and ethics

Search

Navigation