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Homogeneous Lie algebras

Published online by Cambridge University Press:  17 April 2009

S. Świerczkowski
S. Świerczkowski
Affiliation:
Institute of Advanced Studies, Australian National University, Canberra, ACT.
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Abstract

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It is shown that the automorphism group of a real Lie algebra operates transitively on the set of its one-dimensional subspaces iff the Lie algebra is abelian, or isomorphic to the algebra of skew-symmetric 3 × 3 real matrices. This allows to conclude that R, S0(2), S0(3) and Spin(3) are the only connected Lie groups such that: (1) the conjugates of every connected set containing e cover a neighbourhood of e, (2) every point sufficiently close to e lies on exactly one 1-parameter subgroup.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 4 , Issue 3 , June 1971 , pp. 349 - 353
Copyright
Copyright © Australian Mathematical Society 1971

References

[1]Helgason, Sigurđur, Differential geometry and symmetric spaces (Academic Press, New York, London, 1962).Google Scholar
[2]Pontrjagin, L.S., Topologisahe Gruppen. 2 (B.G. Teubner Verlagsgesellschaft, Leipzig, 1958).Google Scholar