The Cartan decomposition is a decomposition of a semisimple Lie group

Lie group

In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...

 or Lie algebra

Lie algebra

In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...

, 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 be a real semisimple Lie algebra

Lie algebra

In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...

 and let be its Killing form

Killing form

In mathematics, the Killing form, named after Wilhelm Killing, is a symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras...

. An involution on is a Lie algebra automorphism

Automorphism

In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism...

  of whose square is equal to the identity. Such an involution is called a Cartan involution on if is a positive definite bilinear form.

Two involutions and 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 is defined by , where denotes the transpose matrix of .

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

• Let be the complexification of a real semisimple Lie algebra , then complex conjugation on is an involution on . This is the Cartan involution on if and only if is the Lie algebra of a compact Lie group.

• The following maps are involutions of the Lie algebra of the special unitary group SU(n):

• • the identity involution , which is the unique Cartan involution in this case;

• • which on is also the complex conjugation;

• • if is odd, . These are all equivalent, but not equivalent to the identity involution (because the matrix does not belong to .)

• • if is even, we also have

Cartan pairs

Let be an involution on a Lie algebra . Since , the linear map has the two eigenvalues . Let and be the corresponding eigenspaces, then . Since is a Lie algebra automorphism, eigenvalues are multiplicative. It follows that

, , and .

Thus is a Lie subalgebra, while any subalgebra of is commutative.

Conversely, a decomposition with these extra properties determines an involution on that is on and on .

Such a pair is also called a Cartan pair of .

The decomposition associated to a Cartan involution is called a Cartan decomposition of . The special feature of a Cartan decomposition is that the Killing form is negative definite on and positive definite on . Furthermore, and are orthogonal complements of each other with respect to the Killing form on .

Cartan decomposition on the Lie group level

Let be a semisimple Lie group

Lie group

In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...

 and its Lie algebra

Lie algebra

In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...

. Let be a Cartan involution on and let be the resulting Cartan pair. Let be the analytic subgroup of with Lie algebra . Then

• There is a Lie group automorphism with differential that satisfies .
• The subgroup of elements fixed by is ; in particular, is a closed subgroup.
• The mapping given by is a diffeomorphism.
• The subgroup contains the center of , and is compact modulo center, that is, is compact.
• The subgroup is the maximal subgroup of that contains the center and is compact modulo center.

The automorphism is also called global Cartan involution, and the diffeomorphism is called global Cartan decomposition.

For the general linear group, we get as the Cartan involution.

Relation to polar decomposition

Consider with the Cartan involution . Then is the real Lie algebra of skew-symmetric matrices, so that , while is the subspace of symmetric matrices. Thus the exponential map is a diffeomorphism from 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.