Cartan decomposition

Let ${\displaystyle {\mathfrak {g}}}$  be a real semisimple Lie algebra and let ${\displaystyle B(\cdot ,\cdot )}$  be its Killing form. An involution on ${\displaystyle {\mathfrak {g}}}$  is a Lie algebra automorphism ${\displaystyle \theta }$  of ${\displaystyle {\mathfrak {g}}}$  whose square is equal to the identity. Such an involution is called a Cartan involution on ${\displaystyle {\mathfrak {g}}}$  if ${\displaystyle B_{\theta }(X,Y):=-B(X,\theta Y)}$  is a positive definite bilinear form.

Two involutions ${\displaystyle \theta _{1}}$  and ${\displaystyle \theta _{2}}$  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.

• A Cartan involution on ${\displaystyle {\mathfrak {sl}}_{n}(\mathbb {R} )}$  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 a compact semisimple Lie group.
• Let ${\displaystyle {\mathfrak {g}}}$  be the complexification 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 ${\displaystyle {\mathfrak {su}}(n)}$  of the special unitary group SU(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 ${\displaystyle {\mathfrak {su}}(2)}$ .
  3. If ${\displaystyle n=p+q}$  is odd, ${\displaystyle \theta _{3}(X)={\begin{pmatrix}I_{p}&0\\0&-I_{q}\end{pmatrix}}X{\begin{pmatrix}I_{p}&0\\0&-I_{q}\end{pmatrix}}}$ . The involutions (1), (2) and (3) are equivalent, but not equivalent to the identity involution since ${\displaystyle {\begin{pmatrix}I_{p}&0\\0&-I_{q}\end{pmatrix}}\notin {\mathfrak {su}}(n)}$ .
  4. If ${\displaystyle n=2m}$  is even, there is also ${\displaystyle \theta _{4}(X)={\begin{pmatrix}0&I_{m}\\-I_{m}&0\end{pmatrix}}X^{T}{\begin{pmatrix}0&I_{m}\\-I_{m}&0\end{pmatrix}}}$ .

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 ${\displaystyle H^{*}({\mathfrak {g}},{\mathfrak {k}})}$ .

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 ${\displaystyle {\mathfrak {g}}}$ .

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

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

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

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.

Consider ${\displaystyle {\mathfrak {gl}}_{n}(\mathbb {R} )}$  with the Cartan involution ${\displaystyle \theta (X)=-X^{T}}$ .^{[clarification needed]} Then ${\displaystyle {\mathfrak {k}}={\mathfrak {so}}_{n}(\mathbb {R} )}$  is the real Lie algebra of skew-symmetric matrices, so that ${\displaystyle K=\mathrm {SO} (n)}$ , 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.

• Lie group decompositions

• 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.