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Actions of Compact Lie Groups

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Part of the Encyclopaedia of Mathematical Sciences book series (EMS,volume 20)

Abstract

The theory of compact Lie transformation groups is a large and well researched area of mathematics, in which topological methods are widely used and which requires a separate survey. In this chapter we only touch upon certain problems of this theory. In §1 we shall survey the classical results of Gleason, Montgomery, Mostow, Palais, Samelson and other authors concerning the structure of of orbits of actions of compact Lie groups. We emphasize the differentiable aspects of the theory and a number of results can be formulated in a more general setting — for proper actions of not necessarily compact Lie groups. §2 is devoted to properties of differentiable invariants and almost-invariants of compact Lie groups. In §3 we consider complexifications of homogeneous spaces of compact Lie groups and factorizations of reductive algebraic groups. Surveys of many questions not considered here can be found in (Bourbaki 1982), (Bredon 1972), (Hsiang 1975), (Jänich 1968), (Palais 1960), (Schultz 1984).

Keywords

  • Homogeneous Space
  • Algebraic Group
  • Orbit Type
  • Linear Algebraic Group
  • Principal Orbit

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© 1993 Springer-Verlag Berlin Heidelberg

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Onishchik, A.L. (1993). Actions of Compact Lie Groups. In: Onishchik, A.L. (eds) Lie Groups and Lie Algebras I. Encyclopaedia of Mathematical Sciences, vol 20. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-57999-8_9

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  • DOI: https://doi.org/10.1007/978-3-642-57999-8_9

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-61222-3

  • Online ISBN: 978-3-642-57999-8

  • eBook Packages: Springer Book Archive