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| Cartan decomposition | |
|---|---|
| Name | Cartan decomposition |
| Field | Cartan theory, Lie groups, Lie algebras, Riemannian geometry, Representation theory |
| Introduced | 20th century |
| Key people | Élie Cartan, Hermann Weyl, Élie Cartan , Harish-Chandra, Claude Chevalley |
| Related concepts | Cartan involution, Iwasawa decomposition, Root system, Weyl group, Cartan subalgebra |
Cartan decomposition The Cartan decomposition is a structural decomposition in the theory of Lie groups and Lie algebras that expresses a semisimple object as the sum or product of symmetric pieces determined by a Cartan involution. It links foundational contributions of Élie Cartan with later developments by Hermann Weyl, Harish-Chandra, and Claude Chevalley and plays a central role in the study of Riemannian symmetric spaces, representation theory, and classification of semisimple Lie algebras.
Cartan decomposition originated in work by Élie Cartan on symmetric spaces and was systematized through interactions with results of Hermann Weyl and Harish-Chandra. It connects structures such as Cartan involution, Cartan subalgebra, root system, and the Weyl group and complements decompositions like the Iwasawa decomposition. Fundamental examples arise from classical groups such as SO(n), SU(n), and Sp(n), and from noncompact real forms classified in the work of Élie Cartan and later catalogued by Armand Borel and N. Jacobson.
A Cartan involution is an involutive automorphism introduced by Élie Cartan that yields a decomposition of a real semisimple Lie algebra g into +1 and −1 eigenspaces. For a real form g of a complex semisimple Lie algebra g_C, one obtains g = k ⊕ p with k the fixed-point algebra and p the −1 eigenspace; this is the algebraic Cartan decomposition. The decomposition satisfies bracket relations [k,k] ⊂ k, [k,p] ⊂ p, [p,p] ⊂ k and is compatible with the Killing form in the sense that the Killing form is negative definite on k and positive definite on p after suitable sign choices. The corresponding global statement for a connected real semisimple Lie group G with maximal compact subgroup K (obtained via integration of k) is G = K exp(p), linking to structures studied by Élie Cartan, Harish-Chandra, and I. M. Gelfand.
For a real semisimple Lie algebra g, choose a Cartan involution θ (motivated by Élie Cartan). The eigenspace decomposition g = k ⊕ p yields a maximal compact subalgebra k and a complementary subspace p, with the pair (g,k) forming a symmetric pair in the classification of Riemannian symmetric spaces by Élie Cartan. A Cartan subalgebra a of p leads to a restricted root system Σ in a*, and the associated multiplicities and Weyl group structure (as in work by Harish-Chandra and H. Weyl) govern the representation-theoretic and harmonic-analytic consequences. The algebraic Cartan decomposition interacts with the Cartan subalgebra theory for complexifications g_C and with the classification of real forms due to Élie Cartan and later tables by Armand Borel.
At the group level, for a connected semisimple real Lie group G with Cartan involution θ, the fixed-point subgroup K is a maximal compact subgroup (classic results attributable to Élie Cartan and formalized by Harish-Chandra). The Cartan decomposition states G = K exp( p ), and more precisely G = K A K for A = exp(a) when a is a maximal abelian subspace of p; this double-coset description parallels the Bruhat decomposition and refines the Iwasawa decomposition G = K A N. These decompositions are central in harmonic analysis on G developed by Harish-Chandra and later by Harish-Chandra’s school and influence Plancherel formulae and spherical function theory studied by H. Weyl and Elie Cartan.
Classical examples include the decomposition of g = so(n,1) with maximal compact k = so(n) arising from the pseudo-orthogonal groups SO(n,1), the noncompact real forms of SL(n,C) like SL(n,R), and indefinite unitary groups such as U(p,q). Exceptional real forms classified by Élie Cartan produce Cartan decompositions for exceptional groups like G2, F4, E6, E7, and E8 over R. The classification of symmetric pairs (g,k) and of real semisimple Lie algebras by signature and Satake diagrams was advanced by Élie Cartan and later catalogued by Armand Borel and Nathan Jacobson.
Cartan decomposition underlies the classification of irreducible unitary representations via highest-weight theory linked to Hermann Weyl and the work of Harish-Chandra on characters and discrete series. In Riemannian geometry it identifies global structures of Riemannian symmetric spaces of noncompact type and facilitates harmonic analysis of eigenfunctions on locally symmetric spaces studied by Atle Selberg and Harish-Chandra. It also appears in geometric representation theory contexts involving Borel subgroups, Cartan subalgebra actions, and in analytic approaches to automorphic forms developed by Atle Selberg and Robert Langlands.
Key technical results use properties of the Killing form, existence and uniqueness (up to conjugacy) of Cartan involutions (proved in Cartan’s work and refined by Harish-Chandra), and the structure theory of root systems and Weyl chambers detailed by Hermann Weyl and Claude Chevalley. Proofs of G = K exp(p) and G = K A K rely on polar decomposition arguments and integration of Lie algebra data to group-level statements, and are standard in texts by Armand Borel, N. Jacobson, and expositions following Élie Cartan.
Category:Lie groups Category:Lie algebras Category:Riemannian geometry