# Contents

## Idea

Theorem (Chevalley)

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 the complex Lie group $G(\mathbb{C})$ is the complexification of $G(\mathbb{R})$.

## 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)