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. 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 formula_1 be a real semisimple Lie algebra and let formula_2 be its Killing form. An involution on formula_1 is a Lie algebra automorphism formula_4 of formula_1 whose square is equal to the identity. Such an involution is called a "Cartan involution" on formula_1 if formula_7 is a positive definite bilinear form.
Two involutions formula_8 and formula_9 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.
Cartan pairs.
Let formula_4 be an involution on a Lie algebra formula_1. Since formula_35, the linear map formula_4 has the two eigenvalues formula_37. If formula_38 and formula_39 denote the eigenspaces corresponding to +1 and -1, respectively, then formula_40. Since formula_4 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 formula_38 is a Lie subalgebra, while any subalgebra of formula_39 is commutative.
Conversely, a decomposition formula_40 with these extra properties determines an involution formula_4 on formula_1 that is formula_50 on formula_38 and formula_52 on formula_39.
Such a pair formula_54 is also called a "Cartan pair" of formula_1, 
and formula_56 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 formula_57.
The decomposition formula_40 associated to a Cartan involution is called a "Cartan decomposition" of formula_1. The special feature of a Cartan decomposition is that the Killing form is negative definite on formula_38 and positive definite on formula_39. Furthermore, formula_38 and formula_39 are orthogonal complements of each other with respect to the Killing form on formula_1.
Cartan decomposition on the Lie group level.
Let formula_65 be a non-compact semisimple Lie group and formula_1 its Lie algebra. Let formula_4 be a Cartan involution on formula_1 and let formula_69 be the resulting Cartan pair. Let formula_70 be the analytic subgroup of formula_65 with Lie algebra formula_38. Then:
The automorphism formula_73 is also called the "global Cartan involution", and the diffeomorphism formula_79 is called the "global Cartan decomposition". If we write formula_85
this says that the product map formula_86 is a diffeomorphism so formula_87.
For the general linear group, formula_88 is a Cartan involution.
A refinement of the Cartan decomposition for symmetric spaces of compact or noncompact type states that the maximal Abelian subalgebras formula_89 in formula_39 are unique up to conjugation by formula_70. Moreover,
where formula_93.
In the compact and noncompact case the global Cartan decomposition thus implies
Geometrically the image of the subgroup formula_95 in formula_96 is a totally geodesic submanifold.
Relation to polar decomposition.
Consider formula_97 with the Cartan involution formula_11. Then formula_99 is the real Lie algebra of skew-symmetric matrices, so that formula_100, while formula_39 is the subspace of symmetric matrices. Thus the exponential map is a diffeomorphism from formula_39 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.