Cartan decomposition of a real reductive Lie algebra

Definition

Given a Cartan involution $\theta$ of a real reductive Lie algebra $\mathfrak g$ , its Cartan decomposition is the vector-space direct sum

\mathfrak g=\mathfrak k\oplus\mathfrak p,\qquad \mathfrak k=\ker(\theta-1),\quad \mathfrak p=\ker(\theta+1).

Because $\theta$ is a Lie algebra automorphism, the brackets satisfy

[\mathfrak k,\mathfrak k]\subseteq\mathfrak k,\qquad [\mathfrak k,\mathfrak p]\subseteq\mathfrak p,\qquad [\mathfrak p,\mathfrak p]\subseteq\mathfrak k.

Thus $\mathfrak k$ is a Lie subalgebra, whereas $\mathfrak p$ is generally not. The decomposition is orthogonal for the positive inner product associated with $\theta$ .

Compact and noncompact directions

The subalgebra $\mathfrak k$ is maximally compactly embedded and integrates, under the usual group hypotheses, to a maximal compact subgroup $K$ . The space $\mathfrak p$ models the tangent space at the base point of the homogeneous space $G/K$ . The bracket inclusions are the infinitesimal algebraic signature of its symmetric-space geometry Helgason, Chapter V.

Example

For $\mathfrak g=\mathfrak{gl}_n(\mathbb R)$ and $\theta(X)=-X^{\mathsf T}$ ,

\mathfrak k=\mathfrak{so}(n),\qquad \mathfrak p=\{X:X^{\mathsf T}=X\}.

Exponentiating $\mathfrak p$ produces the positive-definite symmetric matrices. The resulting group-level statement is the familiar polar decomposition and is a model for the global Cartan decomposition.

Terminological distinction

This $\mathfrak k\oplus\mathfrak p$ decomposition should not be confused with a root-space decomposition relative to a Cartan subalgebra. A maximal abelian subspace $\mathfrak a\subseteq\mathfrak p$ instead gives the restricted root system, which records how the noncompact directions decompose under $\operatorname{ad}\mathfrak a$ .

References

A. W. Knapp, Lie Groups Beyond an Introduction, 2nd ed., Birkhäuser, 2002. Publisher record. Relevant: Chapter VI, §2.
S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, American Mathematical Society, 2001. DOI record. Relevant: Chapter V on symmetric decompositions.