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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJun 20th 2014
   • (edited Jun 20th 2014)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI gather (via [this nice MO comment](http://ncatlab.org/nlab/show/relation+between+compact+Lie+groups+and+reductive+algebraic+groups#Conrad10)) that > The [[functor]] that takes linear [[algebraic groups]] $G$ to their $\mathbb{R}$-points $G(\mathbb{R})$ 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 $G$ as in this equivalence, then then [[complex Lie group]] $G(\mathbb{C})$ is the [[complexification]] of $G(\mathbb{R})$. I have a gap in my education here and would like to fill it. What's a good source that discusses this statement a bit more? And which one of Chevalley's articles is this result originally due?

   I gather (via this nice MO comment) that

   The functor that takes linear algebraic groups $G$ to their $\mathbb{R}$-points $G(\mathbb{R})$ 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 $G$ as in this equivalence, then then complex Lie group $G(\mathbb{C})$ is the complexification of $G(\mathbb{R})$.

   I have a gap in my education here and would like to fill it. What’s a good source that discusses this statement a bit more? And which one of Chevalley’s articles is this result originally due?