Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-20T04:36:56.815Z Has data issue: false hasContentIssue false
  • English
  • Français

Varieties of topological groups generated by Lie groups

Published online by Cambridge University Press:  20 January 2009

Su-Shing Chen  and
Sidney A. Morris
Su-Shing Chen
Affiliation:
University of Florida, Gainesville, Florida, U.S.A.
Sidney A. Morris
Affiliation:
The University of New South Wales, Kensington, New South Wales, Australia
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.

Varieties of topological groups have been investigated in several papers ((2) and (10)-(13)). In this note we investigate the varieties generated by classical Lie groups. In particular we show results of which the following is indicative: The variety generated by the unitary group U(n) contains U(m) if and only if m ≦ n. En route we introduce the notion of a variety of topological Lie algebras which provides a convenient setting in which to answer our questions.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 18 , Issue 1 , June 1972 , pp. 49 - 53
Copyright
Copyright © Edinburgh Mathematical Society 1972

References

REFERENCES

(1) Balcerzyk, S. and Mycielski, Jan, On the existence of free subgroups in topological groups, Fundamenta Mathematicae 44 (1957), 303–308.CrossRefGoogle Scholar
(2) Brooks, M. S., Morris, Sidney A. and Saxon, Stephen A., Generating varieties of topological groups, Proc. Edinburgh Math. Soc. To appear.Google Scholar
(3) Diestel, Joseph, Morris, Sidney A. and Saxon, Stephen A., Varieties of locally convex topological vector spaces, Bull. Amer. Math. Soc. 77 (1971), 799–803.CrossRefGoogle Scholar
(4) Diestel, Joseph, Morris, Sidney A. and Saxon, Stephen A., Varieties of linear topological spaces, Trans. Amer. Math. Soc. To appear.Google Scholar
(5) Helgason, S., Differential Geometry and Symmetric Spaces (Academic Press, New York, 1962).Google Scholar
(6) Hewitt, E. and Ross, K. A., Abstract Harmonic Analysis II (Springer-Verlag, Berlin, Heidelberg, 1970).Google Scholar
(7) Hochschild, G., The Structure of Lie Groups (Holden-Day Inc., San Francisco, London, Amsterdam, 1965).Google Scholar
(8) Hofmann, K. H., Introduction to the Theory of Compact Groups I (Tulane University Lecture notes, 1968).Google Scholar
(9) Montgomery, D. and Zippin, L., Topological Transformation Groups (Inter-science tracts in pure and applied mathematics I, New York, 1955).Google Scholar
(10) Morris, Sidney A., Varieties of topological groups, Bull. Australian Math. Soc. 1 (1969), 145–160.CrossRefGoogle Scholar
(11) Morris, Sidney A., Varieties of topological groups II, Bull. Australian Math. Soc. 2 (1970), 1–13.CrossRefGoogle Scholar
(12) Morris, Sidney A., Varieties of topological groups II, Bull. Australian Math. Soc. 2 (1970), 165–178.CrossRefGoogle Scholar
(13) Morris, Sidney A., Locally compact Abelian groups and the variety of topological groups generated by the reals, Proc. Amer. Math. Soc. To appear.Google Scholar
(14) Neumann, Hanna, Varieties of Groups (Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 37, Springer-Verlag, Berlin, Heidelberg, 1967).Google Scholar