Cartan decomposition

In mathematics, 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.

Its history can be traced to the 1880s work of Élie Cartan and Wilhelm Killing.

be a real semisimple Lie algebra and let

whose square is equal to the identity.

is a positive definite bilinear form.

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.

be an involution on a Lie algebra

denote the eigenspaces corresponding to +1 and -1, respectively, then

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.

with these extra properties determines an involution

is also called a Cartan pair of

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

The special feature of a Cartan decomposition is that the Killing form is negative definite on

are orthogonal complements of each other with respect to the Killing form on

be a non-compact semisimple Lie group and

be the resulting Cartan pair.

is also called the global Cartan involution, and the diffeomorphism

is called the global Cartan decomposition.

For the general linear group,

[clarification needed] A refinement of the Cartan decomposition for symmetric spaces of compact or noncompact type states that the maximal Abelian subalgebras

In the compact and noncompact case the global Cartan decomposition thus implies Geometrically the image of the subgroup

is a totally geodesic submanifold.

is the real Lie algebra of skew-symmetric matrices, so that

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.

The polar decomposition of an invertible matrix is unique.