Iwasawa decomposition
In mathematics, the Iwasawa decomposition 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. 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.
and the Iwasawa decomposition of G is
meaning there is an analytic diffeomorphism from the manifold to the Lie group, sending.
The dimension of A is equal to the real rank of G.
Iwasawa decompositions also hold for some disconnected semisimple groups G, where K becomes a 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, 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 is in terms of
For the symplectic group G = Sp, 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.