In mathematics, the Iwasawa decomposition (aka KAN from its expression) of a semisimple Lie group generalises the way a square real matrix can be written as a product of an orthogonal matrix and an upper triangular matrix (QR decomposition, a consequence of Gram–Schmidt orthogonalization). It is named after Kenkichi Iwasawa, the Japanese mathematician who developed this method.
Definition G is a connected semisimple real Lie group. is the Lie algebra of G is the complexification of . θ is a Cartan involution of is the corresponding Cartan decomposition is a maximal abelian subalgebra of Σ is the set of restricted roots of , corresponding to eigenvalues of acting on . Σ+ is a choice of positive roots of Σ is a nilpotent Lie algebra given as the sum of the root spaces of Σ+ K, A, N, are the Lie subgroups of G generated by and .
Then the Iwasawa decomposition of is
and the Iwasawa decomposition of G is
meaning there is an analytic diffeomorphism (but not a group homomorphism) from the manifold to the Lie group , sending .
The dimension of A (or equivalently of ) is equal to the real rank of G.
Iwasawa decompositions also hold for some disconnected semisimple groups G, where K becomes a (disconnected) maximal compact subgroup provided the center of G is finite.
The restricted root space decomposition is
where is the centralizer of in and is the root space. The number is called the multiplicity of .
Examples If G=SLn(R), then we can take K to be the orthogonal matrices, A to be the positive diagonal matrices with determinant 1, and N to be the unipotent group consisting of upper triangular matrices with 1s on the diagonal.
For the case of n = 2, the Iwasawa decomposition of G = SL(2, R) is in terms of
For the symplectic group G = Sp(2n, R), a possible Iwasawa decomposition is in terms of
Obtaining the matrices appearing in the decomposition above can be reduced to the calculation of matrix square roots, matrix inverses and performing a QR decomposition.
Non-Archimedean Iwasawa decomposition There is an analog to the above Iwasawa decomposition for a non-Archimedean field : In this case, the group can be written as a product of the subgroup of upper-triangular matrices and the (maximal compact) subgroup , where is the ring of integers of ., Prop. 4.5.2
See also Lie group decompositions Root system of a semi-simple Lie algebra
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