nLab
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