In mathematics, the Iwasawa decomposition KAN 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 (a consequence of Gram-Schmidt orthogonalization). It is named after Kenkichi Iwasawa, the Japanese mathematician who developed this method.
- 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
The dimension of A (or equivalently of ) is called 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 .
If G=GLn(R), then we can take K to be the orthogonal matrices, A to be the positive diagonal matrices, and N to be the unipotent group consisting of upper triangular matrices with 1s on the diagonal.
Non-Archimedean Iwasawa decomposition
There is an analogon 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 . 
- Fedenko, A.S.; Shtern, A.I. (2001), "I/i053060", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- A. W. Knapp, Structure theory of semisimple Lie groups, in ISBN 0-8218-0609-2: Representation Theory and Automorphic Forms: Instructional Conference, International Centre for Mathematical Sciences, March 1996, Edinburgh, Scotland (Proceedings of Symposia in Pure Mathematics) by T. N. Bailey (Editor), Anthony W. Knapp (Editor)
- Iwasawa, Kenkichi: On some types of topological groups. Annals of Mathematics (2) 50, (1949), 507–558.
- Bump, Automorphic Forms and Representations, Prop. 4.5.2