nLab Cartan subalgebra

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 $Nor_\mathfrak{g}(\mathfrak{h})=\mathfrak{h}$. 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

$$T \hookrightarrow G \,.$$

Related concepts

• Cartan subgroup

• rank of a Lie group

References

• Nathan Jacobson, Ch. III of: Lie Algebras, Dover Books (1979) [ISBN:]

• Wikipedia, Cartan subalgebra