Editing Cartan decomposition

{{Short description|Generalized matrix decomposition for Lie groups and Lie algebras}}
In mathematics, 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]].<ref>{{harvnb|Kleiner|2007}}</ref> 

== Cartan involutions on Lie algebras ==

Let <math>\mathfrak{g}</math> be a real [[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 ===

* 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. 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 or, equivalently, if and only if <math>\mathfrak{g}</math> is the Lie algebra of a [[Compact Lie group|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_1(X) = X</math>, which is the unique Cartan involution in this case.
*# [[Complex conjugation]], expressible as <math>\theta_2 (X) = - X^T</math> on <math>\mathfrak{su}(2)</math>.
*# If <math>n = p+q</math> is odd, <math>\theta_3 (X) = \begin{pmatrix} I_p & 0 \\ 0 & -I_q \end{pmatrix} X \begin{pmatrix} I_p & 0 \\ 0 & -I_q \end{pmatrix}</math>. The involutions (1), (2) and (3) are equivalent, but not equivalent to the identity involution since <math>\begin{pmatrix} I_p & 0 \\ 0 & -I_q \end{pmatrix} \notin \mathfrak {{su}}(n)</math>.
*# If <math>n = 2m</math> is even, there is also <math>\theta_4 (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>.  If <math>\mathfrak{k}</math> and <math>\mathfrak{p}</math> denote the eigenspaces corresponding to +1 and -1, respectively, then <math>\mathfrak{g} = \mathfrak{k}\oplus\mathfrak{p}</math>.  Since <math>\theta</math> 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

: <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}\oplus\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>, 
and <math>(\mathfrak{g},\mathfrak{k})</math> is called a ''symmetric pair''. This notion of a Cartan pair here is not to be confused with the [[Cartan pair|distinct notion]] involving the relative Lie algebra cohomology <math>H^*(\mathfrak{g},\mathfrak{k})</math>.

The decomposition <math>\mathfrak{g} = \mathfrak{k}\oplus\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 non-compact 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> at the identity 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> is a maximal compact subgroup of <math>G</math>, whenever the center of G is finite.

The automorphism <math>\Theta</math> is also called the ''global Cartan involution'', and the diffeomorphism <math>K\times\mathfrak{p} \rightarrow G</math> is called the ''global Cartan decomposition''. If we write <math>P=\mathrm{exp}(\mathfrak{p})\subset G</math>
this says that the product map <math>K\times P \rightarrow G</math> is a diffeomorphism so <math>G=KP</math>.

For the general linear group, <math> X \mapsto (X^{-1})^T </math> is a Cartan involution.{{clarify|reason=Very first talk page section points out that the Killing-form definition won't work for GL(n). A later talk-page section (titled "is this true?") points out that the transpose won't work for the analytic-subgroup claims above. So, for GL(n), is "any old involution" a Cartan involution? See "Inconsistency!" on talk page.|date=October 2020}}

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 <math>K</math>. Moreover,

:<math>\displaystyle{\mathfrak{p}= \bigcup_{k\in K} \mathrm{Ad}\, k \cdot \mathfrak{a}}
\qquad\text{and}\qquad
\displaystyle{P= \bigcup_{k\in K} \mathrm{Ad}\, k \cdot A}
</math> 
where <math>A = e^\mathfrak{a}</math>.

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

:<math>G = KP = KAK,</math>

Geometrically the image of the subgroup <math>A</math> in <math>G/K</math> is 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>.{{clarify|reason=The very first comment on the talk page points out that the Killing form definition of the Cartan involution doesn't work for gl(n), so what is going on, here? How'd you get this? This is also inconsistent with the analytic-group comment on the talk page (see the "is this true?" section) |date=October 2020}} 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. The polar decomposition of an invertible matrix is unique.

== See also ==

* [[Lie group decompositions]]

== Notes ==
{{reflist}}

== References ==
{{more footnotes|date=March 2016}}
*{{citation|first=Sigurdur|last= Helgason|author-link=Sigurdur Helgason (mathematician)|title=Differential geometry, Lie groups, and symmetric spaces|year=1978|publisher=Academic Press|series=Pure and Applied Mathematics| volume=80|isbn=0-8218-2848-7|mr=0514561}}
*{{cite book|last=Kleiner|first=Israel|editor1-first=Israel|editor1-last=Kleiner|title=A History of Abstract Algebra|year=2007|isbn=978-0817646844|doi=10.1007/978-0-8176-4685-1|publisher=Birkhäuser|location=Boston, MA|mr=2347309}}
*{{cite book|last=Knapp|first=Anthony W.|author-link=Anthony W. Knapp|title=Lie groups beyond an introduction|year=2005|orig-year=1996|edition=2nd|isbn=0-8176-4259-5|publisher=Birkhäuser|location=Boston, MA|series=Progress in Mathematics|volume=140|mr=1920389}}

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