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 (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
meaning there is an analytic diffeomorphism (but not a group homomorphism) from the manifold to the Lie group , sending .
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=SLn(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.
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 : 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 .
- Iwasawa, Kenkichi (1949). "On Some Types of Topological Groups". Annals of Mathematics. 50 (3): 507–558. doi:10.2307/1969548. JSTOR 1969548.
- Bump, Daniel (1997), Automorphic forms and representations, Cambridge: Cambridge University Press, doi:10.1017/CBO9780511609572, ISBN 0-521-55098-X, Prop. 4.5.2
- Fedenko, A.S.; Shtern, A.I. (2001) , "Iwasawa decomposition", in Hazewinkel, Michiel (ed.), Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
- Knapp, A. W. (2002). Lie groups beyond an introduction (2nd ed.). ISBN 9780817642594.