Contents

# 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 $G(\mathbb{R})$.