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. It generalizes the polar decomposition or singular value decomposition of matrices. Its history can be traced to the 1880s work of Élie Cartan and Wilhelm Killing.

Cartan involutions on Lie algebras

Let be a real semisimple Lie algebra and let be its Killing form. An involution on is a Lie algebra automorphism \theta 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 \theta_1 and \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.

Examples

Cartan pairs

Let \theta be an involution on a Lie algebra. Since \theta^2=1, the linear map \theta has the two eigenvalues \pm1. If and denote the eigenspaces corresponding to +1 and -1, respectively, then. Since \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 Thus is a Lie subalgebra, while any subalgebra of is commutative. Conversely, a decomposition with these extra properties determines an involution \theta on that is +1 on and -1 on. Such a pair is also called a Cartan pair of , and 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 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 G be a non-compact semisimple Lie group and its Lie algebra. Let \theta be a Cartan involution on and let be the resulting Cartan pair. Let K be the analytic subgroup of G with Lie algebra. Then: The automorphism \Theta is also called the global Cartan involution, and the diffeomorphism is called the global Cartan decomposition. If we write this says that the product map is a diffeomorphism so G=KP. For the general linear group, is a Cartan involution. A refinement of the Cartan decomposition for symmetric spaces of compact or noncompact type states that the maximal Abelian subalgebras in are unique up to conjugation by K. Moreover, where. In the compact and noncompact case the global Cartan decomposition thus implies Geometrically the image of the subgroup A in G/K is a totally geodesic submanifold.

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. The polar decomposition of an invertible matrix is unique.