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
- G is a connected semisimple real Lie group.
- <math> \mathfrak{g}_0 </math> is the Lie algebra of G
- <math> \mathfrak{g} </math> is the complexification of <math> \mathfrak{g}_0 </math>.
- θ is a Cartan involution of <math> \mathfrak{g}_0 </math>
- <math> \mathfrak{g}_0 = \mathfrak{k}_0 \oplus \mathfrak{p}_0 </math> is the corresponding Cartan decomposition
- <math> \mathfrak{a}_0 </math> is a maximal abelian subalgebra of <math> \mathfrak{p}_0 </math>
- Σ is the set of restricted roots of <math> \mathfrak{a}_0 </math>, corresponding to eigenvalues of <math> \mathfrak{a}_0 </math> acting on <math> \mathfrak{g}_0 </math>.
- Σ<sup>+</sup> is a choice of positive roots of Σ
- <math> \mathfrak{n}_0 </math> is a nilpotent Lie algebra given as the sum of the root spaces of Σ<sup>+</sup>
- K, A, N, are the Lie subgroups of G generated by <math> \mathfrak{k}_0, \mathfrak{a}_0 </math> and <math> \mathfrak{n}_0 </math>.
Then the Iwasawa decomposition of <math> \mathfrak{g}_0 </math> is
:<math>\mathfrak{g}_0 = \mathfrak{k}_0 \oplus \mathfrak{a}_0 \oplus \mathfrak{n}_0</math>
and the Iwasawa decomposition of G is
:<math>G=KAN</math>
meaning there is an analytic diffeomorphism (but not a group homomorphism) from the manifold <math> K \times A \times N </math> to the Lie group <math> G </math>, sending <math> (k,a,n) \mapsto kan </math>.
The dimension of A (or equivalently of <math> \mathfrak{a}_0 </math>) 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
:<math> \mathfrak{g}_0 = \mathfrak{m}_0\oplus\mathfrak{a}_0\oplus_{\lambda\in\Sigma}\mathfrak{g}_{\lambda} </math>
where <math>\mathfrak{m}_0</math> is the centralizer of <math>\mathfrak{a}_0</math> in <math>\mathfrak{k}_0</math> and <math>\mathfrak{g}_{\lambda} = \{X\in\mathfrak{g}_0: [H,X]=\lambda(H)X\;\;\forall H\in\mathfrak{a}_0 \}</math> is the root space. The number
<math>m_{\lambda}= \text{dim}\,\mathfrak{g}_{\lambda}</math> is called the multiplicity of <math>\lambda</math>.
Examples
If G=SL<sub>n</sub>(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
:<math> \mathbf{K} = \left\{
\begin{pmatrix}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{pmatrix} \in SL(2,\mathbb{R}) \ | \ \theta\in\mathbf{R} \right\} \cong SO(2) ,
</math>
:<math>
\mathbf{A} = \left\{
\begin{pmatrix}
r & 0 \\
0 & r^{-1}
\end{pmatrix} \in SL(2,\mathbb{R}) \ | \ r > 0 \right\},
</math>
:<math>
\mathbf{N} = \left\{
\begin{pmatrix}
1 & x \\
0 & 1
\end{pmatrix} \in SL(2,\mathbb{R}) \ | \ x\in\mathbf{R} \right\}.
</math>
For the symplectic group G = Sp(2n, R), a possible Iwasawa decomposition is in terms of
:<math> \mathbf{K} = Sp(2n,\mathbb{R})\cap SO(2n)
= \left\{
\begin{pmatrix}
A & B \\
-B & A
\end{pmatrix} \in Sp(2n,\mathbb{R}) \ | \ A+iB \in U(n) \right\} \cong U(n) ,
</math>
:<math>
\mathbf{A} = \left\{
\begin{pmatrix}
D & 0 \\
0 & D^{-1}
\end{pmatrix} \in Sp(2n,\mathbb{R}) \ | \ D \text{ positive, diagonal} \right\},
</math>
:<math>
\mathbf{N} = \left\{
\begin{pmatrix}
N & M \\
0 & N^{-T}
\end{pmatrix} \in Sp(2n,\mathbb{R}) \ | \ N \text{ upper triangular with diagonal elements = 1},\ NM^T = MN^T \right\}.
</math>
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.
Non-Archimedean Iwasawa decomposition
There is an analog to the above Iwasawa decomposition for a non-Archimedean field <math>F</math>: In this case, the group <math>GL_n(F)</math> can be written as a product of the subgroup of upper-triangular matrices and the (maximal compact) subgroup <math>GL_n(O_F)</math>, where <math>O_F</math> is the ring of integers of <math>F</math>.
See also
- Lie group decompositions
- Root system of a semi-simple Lie algebra