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Stanton86W:

The '''Cartan decomposition''' is a decomposition of a [[Semisimple Lie algebra|semisimple]] [[Lie group]] or [[Lie algebra]], which plays an important role in their structure theory and [[representation theory]]. It generalizes the [[polar decomposition]] or [[singular value decomposition]] of matrices. Its history can be traced to the 1880s work of [[Élie Cartan]] and [[Wilhelm Killing]]. [http://books.google.com/books?id=udj-1UuaOiIC&pg=PA46&dq=history+cartan+decomposition&hl=en&sa=X&ei=aa-wUuCDEMGmkQfNqoHABg&ved=0CDQQ6AEwAQ#v=onepage&q=history%20cartan%20decomposition&f=false]

== Cartan involutions on Lie algebras ==

Let <math>\mathfrak{g}</math> be a real [[Semisimple Lie algebra|semisimple]] [[Lie algebra]] and let <math>B(\cdot,\cdot)</math> be its [[Killing form]]. An [[Involution (mathematics)|involution]] on <math>\mathfrak{g}</math> is a Lie algebra [[automorphism]] <math>\theta</math> of <math>\mathfrak{g}</math> whose square is equal to the identity. Such an involution is called a '''Cartan involution''' on <math>\mathfrak{g}</math> if <math>B_\theta(X,Y) := -B(X,\theta Y)</math> is a [[positive definite bilinear form]].

Two involutions <math>\theta_1</math> and <math>\theta_2</math> are considered equivalent if they differ only by an [[inner automorphism]].

Any real semisimple Lie algebra has a Cartan involution, and any two Cartan involutions are equivalent.

=== Examples ===

{{^|NOTE: Blank lines between items helped source readability, but screwed up list formatting}}
* A Cartan involution on <math>\mathfrak{sl}_n(\mathbb{R})</math> is defined by <math>\theta(X)=-X^T</math>, where <math>X^T</math> denotes the transpose matrix of <math>X</math>.
* The identity map on <math>\mathfrak{g}</math> is an involution, of course. It is the unique Cartan involution of <math>\mathfrak{g}</math> if and only if the Killing form of <math>\mathfrak{g}</math> is negative definite. Equivalently, <math>\mathfrak{g}</math> is the Lie algebra of a compact semisimple Lie group.
* Let <math>\mathfrak{g}</math> be the complexification of a real semisimple Lie algebra <math>\mathfrak{g}_0</math>, then complex conjugation on <math>\mathfrak{g}</math> is an involution on <math>\mathfrak{g}</math>. This is the Cartan involution on <math>\mathfrak{g}</math> if and only if <math>\mathfrak{g}_0</math> is the Lie algebra of a compact Lie group.
* The following maps are involutions of the Lie algebra <math>\mathfrak{su}(n)</math> of the special unitary group [[SU(n)]]:
** the identity involution <math>\theta_0(X) = X</math>, which is the unique Cartan involution in this case;
** <math>\theta_1 (X) = - X^T</math> which on <math>\mathfrak{su}(n)</math> is also the complex conjugation;
** if <math>n = p+q</math> is odd, <math>\theta_2 (X) = \begin{pmatrix} I_p & 0 \\ 0 & -I_q \end{pmatrix} X \begin{pmatrix} I_p & 0 \\ 0 & -I_q \end{pmatrix}</math>. These are all equivalent, but not equivalent to the identity involution (because the matrix <math>\begin{pmatrix} I_p & 0 \\ 0 & -I_q \end{pmatrix}</math> does not belong to <math>\mathfrak{su}(n)</math>.)
** if <math>n = 2m</math> is even, we also have <math>\theta_3 (X) = \begin{pmatrix} 0 & I_m \\ -I_m & 0 \end{pmatrix} X^T \begin{pmatrix} 0 & I_m \\ -I_m & 0 \end{pmatrix}.</math>

== Cartan pairs ==

Let <math>\theta</math> be an involution on a Lie algebra <math>\mathfrak{g}</math>. Since <math>\theta^2=1</math>, the linear map <math>\theta</math> has the two eigenvalues <math>\pm1</math>. Let <math>\mathfrak{k}</math> and <math>\mathfrak{p}</math> be the corresponding eigenspaces, then <math>\mathfrak{g} = \mathfrak{k}+\mathfrak{p}</math>. Since <math>\theta</math> is a Lie algebra automorphism, eigenvalues are multiplicative. It follows that
: <math>[\mathfrak{k}, \mathfrak{k}] \subseteq \mathfrak{k}</math>, <math>[\mathfrak{k}, \mathfrak{p}] \subseteq \mathfrak{p}</math>, and <math>[\mathfrak{p}, \mathfrak{p}] \subseteq \mathfrak{k}</math>.
Thus <math>\mathfrak{k}</math> is a Lie subalgebra, while any subalgebra of <math>\mathfrak{p}</math> is commutative.

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

Such a pair <math>(\mathfrak{k}, \mathfrak{p})</math> is also called a '''Cartan pair''' of <math>\mathfrak{g}</math>.

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

== Cartan decomposition on the Lie group level ==

Let <math>G</math> be a [[Semisimple Lie group|semisimple]] [[Lie group]] and <math>\mathfrak{g}</math> its [[Lie algebra]]. Let <math>\theta</math> be a Cartan involution on <math>\mathfrak{g}</math> and let <math>(\mathfrak{k},\mathfrak{p})</math> be the resulting Cartan pair. Let <math>K</math> be the [[analytic subgroup]] of <math>G</math> with Lie algebra <math>\mathfrak{k}</math>. Then:
* There is a Lie group automorphism <math>\Theta</math> with differential <math>\theta</math> that satisfies <math>\Theta^2=1</math>.
* The subgroup of elements fixed by <math>\Theta</math> is <math>K</math>; in particular, <math>K</math> is a closed subgroup.
* The mapping <math>K\times\mathfrak{p} \rightarrow G</math> given by <math>(k,X) \mapsto k\cdot \mathrm{exp}(X)</math> is a diffeomorphism.
* The subgroup <math>K</math> contains the center <math>Z</math> of <math>G</math>, and <math>K</math> is compact modulo center, that is, <math>K/Z</math> is compact.
* The subgroup <math>K</math> is the maximal subgroup of <math>G</math> that contains the center and is compact modulo center.

The automorphism <math>\Theta</math> is also called '''global Cartan involution''', and the diffeomorphism <math>K\times\mathfrak{p} \rightarrow G</math> is called '''global Cartan decomposition'''.

For the general linear group, we get <math> X \mapsto (X^{-1})^T </math> as the Cartan involution.

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

:<math>\displaystyle{\mathfrak{p}= \bigcup_{k\in K} \mathrm{Ad}\, k \cdot \mathfrak{a}.}</math>

In the compact and noncompact case this Lie algebraic result implies the decomposition

:<math>\displaystyle{G=KAK,}</math>

where ''A'' = exp <math>\mathfrak{a}</math>. Geometrically the image of the subgroup ''A'' in ''G'' / ''K'' ia a [[totally geodesic]] submanifold.

== Relation to polar decomposition ==

Consider <math>\mathfrak{gl}_n(\mathbb{R})</math> with the Cartan involution <math>\theta(X)=-X^T</math>. Then <math>\mathfrak{k}=\mathfrak{so}_n(\mathbb{R})</math> is the real Lie algebra of skew-symmetric matrices, so that <math>K=\mathrm{SO}(n)</math>, while <math>\mathfrak{p}</math> is the subspace of symmetric matrices. Thus the exponential map is a diffeomorphism from <math>\mathfrak{p}</math> onto the space of positive definite matrices. Up to this exponential map, the global Cartan decomposition is the [[polar decomposition]] of a matrix. Notice that the polar decomposition of an invertible matrix is unique.

== See also ==

* [[Lie group decompositions]]

== References ==
* {{citation|first=Sigurdur|last= Helgason|title=Differential geometry, Lie groups, and symmetric spaces|year=1978|publisher=Academic Press|isbn= 0-8218-2848-7}}
*[[A. W. Knapp]], ''Lie groups beyond an introduction'', ISBN 0-8176-4259-5, Birkhäuser.

[[Category:Lie groups]]
[[Category:Lie algebras]]

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