Cartan involution of a real Lie algebra - Imported Knowlpedia Content

Definition

Let $\mathfrak g$ be the Lie algebra of a real reductive Lie group, and choose a nondegenerate invariant symmetric bilinear form $B$ that is negative on the compact central directions and positive on the split central directions. A Cartan involution is a Lie algebra automorphism $\theta:\mathfrak g\to\mathfrak g$ satisfying $\theta^2=1$ such that

B_\theta(X,Y)=-B(X,\theta Y)

is positive definite. On the semisimple derived algebra one may take $B$ to be the Killing form. The qualification concerning the center is necessary because the Killing form of a reductive Lie algebra is degenerate there.

Eigenspaces and geometry

Writing $\mathfrak k$ and $\mathfrak p$ for the $+1$ and $-1$ eigenspaces gives the Cartan decomposition $\mathfrak g=\mathfrak k\oplus\mathfrak p$ . The form $B_\theta$ makes these eigenspaces orthogonal. At group level, $\theta$ integrates to an involutive automorphism whose fixed subgroup is maximal compact under the standard global hypotheses Knapp, Chapter VI, §2.

Existence and uniqueness

Every real reductive Lie algebra admits a Cartan involution. Any two are conjugate by an inner automorphism generated by the noncompact directions. Consequently the compact subalgebra $\mathfrak k$ , although not canonical as a literal subspace without a choice, is unique up to inner automorphism. This is why constructions made from a chosen $\theta$ usually have choice-independent isomorphism types.

Matrix example and terminology

For $\mathfrak{gl}_n(\mathbb R)$ , the map $\theta(X)=-X^{\mathsf T}$ is a Cartan involution. Its fixed algebra is $\mathfrak{so}(n)$ , and its $-1$ -eigenspace consists of symmetric matrices. A Cartan involution is not a Cartan subalgebra and should not be confused with complex conjugation defining an arbitrary real form; it singles out a compact real form after complexification.

References

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