Contents

# Contents

## Definition

Let $\mathbb{K}$ be a field (not necessarily algebraically closed), and let $\mathfrak{g}$ be a Lie algebra. A self-normalizing and nilpotent subalgebra $\mathfrak{h} \leq \mathfrak{g}$ is called a Cartan subalgebra of $\mathfrak{g}$.

By self-normalizing, we mean that $\operatorname{Nor}_\mathfrak{g}(\mathfrak{h})=\mathfrak{h}$. Hence, $\mathfrak{h}$ is not an ideal in any larger subalgebra of $\mathfrak{g}$.

## Examples

In the special case that $G$ is a compact Lie group with Lie algebra $\mathfrak{g}$, a Cartan subalgebra of $\mathfrak{g}$ is a sub-Lie algebra

$\mathfrak{t} \hookrightarrow \mathfrak{g}$

that is the Lie algebra of a maximal torus

$T \hookrightarrow G \,.$

## References

• Nathan Jacobson, Ch. III of: Lie Algebras, Dover Books 1962

• Wikipedia, Cartan subalgebra

Last revised on November 23, 2021 at 18:54:24. See the history of this page for a list of all contributions to it.