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.
[1] 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 root space.
is called the multiplicity of
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 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
is the ring of integers of