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relation between compact Lie groups and reductive algebraic groups

Contents

Idea

Theorem (Chevalley)

The functor that takes linear algebraic groups GG to their \mathbb{R}-points G()G(\mathbb{R}) constitutes an equivalence of categories between compact Lie groups and \mathbb{R}-aniosotropic reductive algebraic groups over \mathbb{R} all whose connected components have \mathbb{R}-points.

For GG as in this equivalence, then the complex Lie group G()G(\mathbb{C}) is the complexification of G()G(\mathbb{R}).

(from (Conrad 10))

References

  • James Milne, Affine Group Schemes; Lie Algebras; Lie Groups; Reductive Groups; Arithmetic Subgroups (web)

    Lie algebras, algebraic groups and Lie groups (pdf)

  • A. L.Onishchik, E. B. Vinberg, Lie groups and algebraic groups, Springer 1990

  • Richard Pink, Compact subgroups of linear algebraic groups (pdf)

  • Brian Conrad, MO comment 2010

Last revised on July 2, 2014 at 00:27:51. See the history of this page for a list of all contributions to it.