Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-24T12:54:37.309Z Has data issue: false hasContentIssue false
  • English
  • Français

Low codimensional embeddings of Sp(n) and Su(n)

Published online by Cambridge University Press:  20 January 2009

G. Walker  and
R. M. W. Wood
G. Walker
Affiliation:
University of ManchesterManchester M13 9PL
R. M. W. Wood
Affiliation:
University of ManchesterManchester M13 9PL
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In [4] Elmer Rees proves that the symplectic group Sp(n) can be smoothly embedded in Euclidean space with codimension 3n, and the unitary group U(n) with codimension n. These are special cases of a result he obtains for a compact connected Lie group G. The general technique is first to embed G/T, where T is a maximal torus, as a maximal orbit of the adjoint representation of G, and then to extendto an embedding of G by using a maximal orbit of a faithful representation of G. In thisnote, we observe that in the cases G = Sp(n) or SU(n) an improved result is obtained byusing the “symplectic torus” S3 x … x S3 in place of T = S1 x … x S1. As in Rees's construction, the normal bundle of the embedding of G is trivial.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 27 , Issue 1 , February 1984 , pp. 25 - 29
Copyright
Copyright © Edinburgh Mathematical Society 1984

References

REFERENCES

1.Borel, A. and Hirzebruch, F., Characteristic classes and homogeneous spaces III, Amer. J. Math. 82 (1960), 491–504.CrossRefGoogle Scholar
2.James, I. M. and Whitehead, J. H. C., The homotopy theory of sphere bundles over spheres (I), Proc. Lord. Math. Soc. (3) 4 (1954), 196–218.CrossRefGoogle Scholar
3.Mann, L. N. and Sicks, J. L., Imbedding of compact Lie groups, Indiana U. Math. J. 20 (1971), 655–665.CrossRefGoogle Scholar
4.Rees, E., Some embeddings of Lie groups in Euclidean space, Mathematika 18 (1971), 152–156.CrossRefGoogle Scholar
5.Toda, H., Composition Methods in Homotopy Groups of Spheres (Ann. Math. Stud. No. 49, Princeton, 1962).Google Scholar