nLab Cartan subalgebra

Contents

Context

\infty-Lie theory

∞-Lie theory (higher geometry)

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Contents

Definition

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

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

Examples

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

𝔱𝔤 \mathfrak{t} \hookrightarrow \mathfrak{g}

that is the Lie algebra of a maximal torus

TG. 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.