Cartan decomposition

The Cartan decomposition is a decomposition of a semisimple Lie group or Lie algebra, which plays an important role in their structure theory and representation theory. It generalizes the polar decomposition of matrices.

Cartan involutions on Lie algebras

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

Two involutions $$heta_1 and $$heta_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.

Examples

* A Cartan involution on $$mathfrak{gl}_n(mathbb{R}) is defined by $$heta(X)=-X^T, where $$X^T denotes the transpose matrix of $$X.

* The identity map on $$mathfrak{g} is an involution, of course. It is the unique Cartan involution of $$mathfrak{g} if and only if the Killing form of $$mathfrak{g} is negative definite. Equivalently, $$mathfrak{g} is the Lie algebra of a compact Lie group.

* Let $$mathfrak{g} be the complexification of a real semisimple Lie algebra $$mathfrak{g}_0, then complex conjugation on $$mathfrak{g} is an involution on $$mathfrak{g}. This is the Cartan involution on $$mathfrak{g} if and only if $$mathfrak{g}_0 is the Lie algebra of a compact Lie group.

* The following maps are involutions of the Lie algebra $$mathfrak{su}(n) of the special unitary group SU(n):

** the identity involution $$heta_0(X) = X, which is the unique Cartan involution in this case;

** $$heta_1 (X) = - X^T;

** $$heta_2 (X) = egin{pmatrix} I_p & 0 \ 0 & -I_q end{pmatrix} X egin{pmatrix} I_p & 0 \ 0 & -I_q end{pmatrix}, where $$p+q = n; these are not equivalent to the identity involution because the matrix $$egin{pmatrix} I_p & 0 \ 0 & -I_q end{pmatrix} does not belong to $$mathfrak{su}(n).

** if $$n = 2m is even, we also have $$heta_3 (X) = egin{pmatrix} 0 & I_m \ -I_m & 0 end{pmatrix} X^T egin{pmatrix} 0 & I_m \ -I_m & 0 end{pmatrix}.

Cartan pairs

Let $$heta be an involution on a Lie algebra $$mathfrak{g}. Since $$heta^2=1, the linear map $$heta has the two eigenvalues $$pm1. Let $$mathfrak{k} and $$mathfrak{p} be the corresponding eigenspaces, then $$mathfrak{g} = mathfrak{k}+mathfrak{p}. Since $$heta is a Lie algebra automorphism, we have: $$ [mathfrak{k}, mathfrak{k}] subseteq mathfrak{k}, $$ [mathfrak{k}, mathfrak{p}] subseteq mathfrak{p}, and $$ [mathfrak{p}, mathfrak{p}] subseteq mathfrak{k}.Thus $$mathfrak{k} is a Lie subalgebra, while $$mathfrak{p} is not.

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

Such a pair $$(mathfrak{k}, mathfrak{p}) is also called a Cartan pair of $$mathfrak{g}.

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

Cartan decomposition on the Lie group level

Let $$G be a semisimple Lie group and $$mathfrak{g} its Lie algebra. Let $$heta be a Cartan involution on $$mathfrak{g} and let $$(mathfrak{k},mathfrak{p}) be the resulting Cartan pair. Let $$K be the analytic subgroup of $$G with Lie algebra $$mathfrak{k}. Then
* There is a Lie group automorphism $$Theta with differential $$heta that satisfies $$Theta^2=1.
* The subgroup of elements fixed by $$Theta is $$K; in particular, $$K is a closed subgroup.
* The mapping $$K imesmathfrak{p} ightarrow G given by $$(k,X) mapsto kcdot mathrm{exp}(X) is a diffeomorphism.
* The subgroup $$K contains the center $$Z of $$G, and $$K is compact modulo center, that is, $$K/Z is compact.
* The subgroup $$K is the maximal subgroup of $$G that contains the center and is compact modulo center.

The automorphism $$Theta is also called global Cartan involution, and the diffeomorphism $$K imesmathfrak{p} ightarrow G is called global Cartan decomposition.

Relation to polar decompostion

Consider $$mathfrak{gl}_n(mathbb{R}) with the Cartain involution $$heta(X)=-X^T. Then $$mathfrak{k}=mathfrak{so}_n(mathbb{R}) is the Lie algebra of skew-symmetric matrices, so that $$K=mathrm{O}(n), while $$mathfrak{p} is the subspace of positive definite matrices. Thus the exponential map is a diffeomorphism from $$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. Notice that the polar decomposition of an invertible matrix is unique.

See also

* Lie group decompositions

References

*A. W. Knapp, "Lie groups beyond an introduction", ISBN 0-8176-4259-5, Birkhäuser.