Cartan decomposition

Cartan pairs

Let ${\displaystyle \theta }$ be an involution on a Lie algebra ${\displaystyle \mathfrak{g}}$. Since ${\displaystyle {\theta }^2=1}$, the linear map ${\displaystyle \theta }$ has the two eigenvalues ${\displaystyle \pm 1}$. If ${\displaystyle \mathfrak{k}}$ and ${\displaystyle \mathfrak{p}}$ denote the eigenspaces corresponding to +1 and -1, respectively, then ${\displaystyle \mathfrak{g}=\mathfrak{k}\oplus \mathfrak{p}}$. Since ${\displaystyle \theta }$ is a Lie algebra automorphism, the Lie bracket of two of its eigenspaces is contained in the eigenspace corresponding to the product of their eigenvalues. It follows that

${\displaystyle [\mathfrak{k},\mathfrak{k}]\subseteq \mathfrak{k}}$, ${\displaystyle [\mathfrak{k},\mathfrak{p}]\subseteq \mathfrak{p}}$, and ${\displaystyle [\mathfrak{p},\mathfrak{p}]\subseteq \mathfrak{k}}$.

Thus ${\displaystyle \mathfrak{k}}$ is a Lie subalgebra, while any subalgebra of ${\displaystyle \mathfrak{p}}$ is commutative.

Conversely, a decomposition ${\displaystyle \mathfrak{g}=\mathfrak{k}\oplus \mathfrak{p}}$ with these extra properties determines an involution ${\displaystyle \theta }$ on ${\displaystyle \mathfrak{g}}$ that is ${\displaystyle +1}$ on ${\displaystyle \mathfrak{k}}$ and ${\displaystyle -1}$ on ${\displaystyle \mathfrak{p}}$.

Such a pair ${\displaystyle (\mathfrak{k},\mathfrak{p})}$ is also called a Cartan pair of ${\displaystyle \mathfrak{g}}$, and ${\displaystyle (\mathfrak{g},\mathfrak{k})}$ is called a symmetric pair. This notion of a Cartan pair here is not to be confused with the distinct notion involving the relative Lie algebra cohomology ${\displaystyle H^{\ast }(\mathfrak{g},\mathfrak{k})}$.

The decomposition ${\displaystyle \mathfrak{g}=\mathfrak{k}\oplus \mathfrak{p}}$ associated to a Cartan involution is called a Cartan decomposition of ${\displaystyle \mathfrak{g}}$. The special feature of a Cartan decomposition is that the Killing form is negative definite on ${\displaystyle \mathfrak{k}}$ and positive definite on ${\displaystyle \mathfrak{p}}$. Furthermore, ${\displaystyle \mathfrak{k}}$ and ${\displaystyle \mathfrak{p}}$ are orthogonal complements of each other with respect to the Killing form on ${\displaystyle \mathfrak{g}}$.

Cartan decomposition on the Lie group level

Let ${\displaystyle G}$ be a non-compact semisimple Lie group and ${\displaystyle \mathfrak{g}}$ its Lie algebra. Let ${\displaystyle \theta }$ be a Cartan involution on ${\displaystyle \mathfrak{g}}$ and let ${\displaystyle (\mathfrak{k},\mathfrak{p})}$ be the resulting Cartan pair. Let ${\displaystyle K}$ be the analytic subgroup of ${\displaystyle G}$ with Lie algebra ${\displaystyle \mathfrak{k}}$. Then:

• There is a Lie group automorphism ${\displaystyle \mathrm{\Theta }}$ with differential ${\displaystyle \theta }$ at the identity that satisfies ${\displaystyle {\mathrm{\Theta }}^2=1}$.
• The subgroup of elements fixed by ${\displaystyle \mathrm{\Theta }}$ is ${\displaystyle K}$; in particular, ${\displaystyle K}$ is a closed subgroup.
• The mapping ${\displaystyle K\times \mathfrak{p}\to G}$ given by ${\displaystyle (k,X)\mapsto k\cdot \mathrm{e}\mathrm{x}\mathrm{p}(X)}$ is a diffeomorphism.
• The subgroup ${\displaystyle K}$ is a maximal compact subgroup of ${\displaystyle G}$, whenever the center of G is finite.

The automorphism ${\displaystyle \mathrm{\Theta }}$ is also called the global Cartan involution, and the diffeomorphism ${\displaystyle K\times \mathfrak{p}\to G}$ is called the global Cartan decomposition. If we write ${\displaystyle P=\mathrm{e}\mathrm{x}\mathrm{p}(\mathfrak{p})\subset G}$ this says that the product map ${\displaystyle K\times P\to G}$ is a diffeomorphism so ${\displaystyle G=KP}$.

For the general linear group, ${\displaystyle X\mapsto (X^{-1}{)}^T}$ is a Cartan involution.

A refinement of the Cartan decomposition for symmetric spaces of compact or noncompact type states that the maximal Abelian subalgebras ${\displaystyle \mathfrak{a}}$ in ${\displaystyle \mathfrak{p}}$ are unique up to conjugation by ${\displaystyle K}$. Moreover,

${\displaystyle {\displaystyle \mathfrak{p}=\bigcup _{k\in K}\mathrm{A}\mathrm{d}k\cdot \mathfrak{a}\text{and}{\displaystyle P=\bigcup _{k\in K}\mathrm{A}\mathrm{d}k\cdot A}}}$

where ${\displaystyle A=e^{\mathfrak{a}}}$.

In the compact and noncompact case the global Cartan decomposition thus implies

${\displaystyle G=KP=KAK,}$

Geometrically the image of the subgroup ${\displaystyle A}$ in ${\displaystyle G/K}$ is a totally geodesic submanifold.

Relation to polar decomposition

Consider ${\displaystyle {\mathfrak{g}\mathfrak{l}}_n(\mathbb{R})}$ with the Cartan involution ${\displaystyle \theta (X)=-X^T}$. Then ${\displaystyle \mathfrak{k}={\mathfrak{s}\mathfrak{o}}_n(\mathbb{R})}$ is the real Lie algebra of skew-symmetric matrices, so that ${\displaystyle K=\mathrm{S}\mathrm{O}(n)}$, while ${\displaystyle \mathfrak{p}}$ is the subspace of symmetric matrices. Thus the exponential map is a diffeomorphism from ${\displaystyle \mathfrak{p}}$ onto the space of positive definite matrices. Up to this exponential map, the global Cartan decomposition is the polar decomposition of a matrix. The polar decomposition of an invertible matrix is unique.