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The Relation Between Lie Groups and Lie Algebras

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

Abstract

The basic method of the theory of Lie groups, which makes it possible to obtain deep results with striking simplicity, consists in reducing questions concerning Lie groups to certain problems of linear algebra. This is done by assigning to every Lie group G its “tangent algebra”g, which to a large extent determines the group G, and to every homomorphism f: GH of Lie groups a homomorphism df: g → h) of their tangent algebras, which to a large extent determines the homomorphism f. In the language of category theory we have a functor from the category of Lie groups into the category of Lie algebras, whose properties are very close to those of an equivalence of categories. In honour of the founder of the theory of Lie groups we will call this functor (following M. M. Postnikov (1982)) the Lie functor.

Keywords

  • Commutator Subgroup
  • Invariant Vector Field
  • Tangent Algebra

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). The Relation Between Lie Groups and Lie Algebras. 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_3

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  • DOI: https://doi.org/10.1007/978-3-642-57999-8_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-61222-3

  • Online ISBN: 978-3-642-57999-8

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