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Transitive Actions

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

Abstract

1.1. Definitions and Examples. Recall that any transitive action of a group G on a set M is isomorphic to an action of this group by left translations on the set of all cosets G/H where H = G x is the stabilizer of any point xM. In the differentiable case (see 1.4 of Chap. 1) G/H is an analytic homogeneous space of the Lie group G and the isomorphism is a diffeomorphism. The G- space G/H is called a group model (or a Klein model) of the homogeneous space M. A group model depends on the choice of the point x so that the subgroup H is defined up to conjugation in G. By means of a group model any property of the homogeneous space M can be expressed in terms of the group G and a subgroup H. We shall now illustrate this method on the simplest examples.

Keywords

  • Homogeneous Space
  • Maximal Compact Subgroup
  • Riemannian Structure
  • Transitive Action
  • Homogeneous Vector Bundle

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). Transitive Actions. 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_8

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  • DOI: https://doi.org/10.1007/978-3-642-57999-8_8

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-61222-3

  • Online ISBN: 978-3-642-57999-8

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