In mathematics, theCartan decomposition is a decomposition of asemisimpleLie group orLie algebra, which plays an important role in their structure theory andrepresentation theory. It generalizes thepolar decomposition orsingular value decomposition of matrices. Its history can be traced to the 1880s work ofÉlie Cartan andWilhelm Killing.^[1]

Let${\displaystyle \mathfrak{g}}$ be a realsemisimple Lie algebra and let${\displaystyle B(\cdot ,\cdot )}$ be itsKilling form. Aninvolution on${\displaystyle \mathfrak{g}}$ is a Lie algebraautomorphism${\displaystyle \theta }$ of${\displaystyle \mathfrak{g}}$ whose square is equal to the identity. Such an involution is called aCartan involution on${\displaystyle \mathfrak{g}}$ if${\displaystyle B_{\theta }(X,Y):=-B(X,\theta Y)}$ is apositive definite bilinear form.

Two involutions${\displaystyle {\theta }_1}$ and${\displaystyle {\theta }_2}$ are considered equivalent if they differ only by aninner automorphism.

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

• A Cartan involution on${\displaystyle {\mathfrak{s}\mathfrak{l}}_n(\mathbb{R})}$ is defined by${\displaystyle \theta (X)=-X^T}$, where${\displaystyle X^T}$ denotes the transpose matrix of${\displaystyle X}$.
• The identity map on${\displaystyle \mathfrak{g}}$ is an involution. It is the unique Cartan involution of${\displaystyle \mathfrak{g}}$ if and only if the Killing form of${\displaystyle \mathfrak{g}}$ is negative definite or, equivalently, if and only if${\displaystyle \mathfrak{g}}$ is the Lie algebra of acompact semisimple Lie group.
• Let${\displaystyle \mathfrak{g}}$ be thecomplexification of a real semisimple Lie algebra${\displaystyle {\mathfrak{g}}_0}$, then complex conjugation on${\displaystyle \mathfrak{g}}$ is an involution on${\displaystyle \mathfrak{g}}$. This is the Cartan involution on${\displaystyle \mathfrak{g}}$ if and only if${\displaystyle {\mathfrak{g}}_0}$ is the Lie algebra of a compact Lie group.
• The following maps are involutions of the Lie algebra${\displaystyle \mathfrak{s}\mathfrak{u}(n)}$ of thespecial unitary groupSU(n):
  1. The identity involution${\displaystyle {\theta }_1(X)=X}$, which is the unique Cartan involution in this case.
  2. Complex conjugation, expressible as${\displaystyle {\theta }_2(X)=-X^T}$ on${\displaystyle \mathfrak{s}\mathfrak{u}(2)}$.
  3. If${\displaystyle n=p+q}$ is odd,. The involutions (1), (2) and (3) are equivalent, but not equivalent to the identity involution since.
  4. If${\displaystyle n=2m}$ is even, there is also.

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 aCartan pair of${\displaystyle \mathfrak{g}}$, and${\displaystyle (\mathfrak{g},\mathfrak{k})}$ is called asymmetric pair. This notion of a Cartan pair here is not to be confused with thedistinct 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 aCartan 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${\displaystyle \mathfrak{g}}$.

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 theanalytic 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 adiffeomorphism.
• 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 theglobal Cartan involution, and the diffeomorphism${\displaystyle K\times \mathfrak{p}\to G}$ is called theglobal 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.^{[clarification needed]}

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 atotally geodesic submanifold.

Consider${\displaystyle {\mathfrak{g}\mathfrak{l}}_n(\mathbb{R})}$ with the Cartan involution${\displaystyle \theta (X)=-X^T}$.^{[clarification needed]} 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 thepolar decomposition of a matrix. The polar decomposition of an invertible matrix is unique.

• Lie group decompositions

This article includes a list ofgeneral references, butit lacks sufficient correspondinginline citations. Please help toimprove this article byintroducing more precise citations.(March 2016) (Learn how and when to remove this message)

• Helgason, Sigurdur (1978),Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, vol. 80, Academic Press,ISBN 0-8218-2848-7,MR 0514561
• Kleiner, Israel (2007). Kleiner, Israel (ed.).A History of Abstract Algebra. Boston, MA: Birkhäuser.doi:10.1007/978-0-8176-4685-1.ISBN 978-0817646844.MR 2347309.
• Knapp, Anthony W. (2005) [1996].Lie groups beyond an introduction. Progress in Mathematics. Vol. 140 (2nd ed.). Boston, MA: Birkhäuser.ISBN 0-8176-4259-5.MR 1920389.

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