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

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 G, 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 \lambda.

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

Non-Archimedean Iwasawa decomposition

There is an analog to the above Iwasawa decomposition for a non-Archimedean field F: In this case, the group GL_n(F) can be written as a product of the subgroup of upper-triangular matrices and the (maximal compact) subgroup GL_n(O_F), where O_F is the ring of integers of F.