Iwasawa decomposition

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.
 * $$ \mathfrak{g}_0 $$ is the Lie algebra of G
 * $$ \mathfrak{g} $$ is the complexification of $$ \mathfrak{g}_0 $$.
 * θ is a Cartan involution of $$ \mathfrak{g}_0 $$
 * $$ \mathfrak{g}_0 = \mathfrak{k}_0 \oplus \mathfrak{p}_0 $$ is the corresponding Cartan decomposition
 * $$ \mathfrak{a}_0 $$ is a maximal abelian subalgebra of $$ \mathfrak{p}_0 $$
 * Σ is the set of restricted roots of $$ \mathfrak{a}_0 $$, corresponding to eigenvalues of $$ \mathfrak{a}_0 $$ acting on $$ \mathfrak{g}_0 $$.
 * Σ+ is a choice of positive roots of Σ
 * $$ \mathfrak{n}_0 $$ 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 $$ \mathfrak{k}_0, \mathfrak{a}_0 $$ and $$ \mathfrak{n}_0 $$.

Then the Iwasawa decomposition of $$ \mathfrak{g}_0 $$ is
 * $$\mathfrak{g}_0 = \mathfrak{k}_0 \oplus \mathfrak{a}_0 \oplus \mathfrak{n}_0$$

and the Iwasawa decomposition of G is
 * $$G=KAN$$

meaning there is an analytic diffeomorphism (but not a group homomorphism) from the manifold $$ K \times A \times N $$ to the Lie group $$ G $$, sending $$ (k,a,n) \mapsto kan $$.

The dimension of A (or equivalently of $$ \mathfrak{a}_0 $$) 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
 * $$ \mathfrak{g}_0 = \mathfrak{m}_0\oplus\mathfrak{a}_0\oplus_{\lambda\in\Sigma}\mathfrak{g}_{\lambda} $$

where $$\mathfrak{m}_0$$ is the centralizer of $$\mathfrak{a}_0$$ in $$\mathfrak{k}_0$$ and $$\mathfrak{g}_{\lambda} = \{X\in\mathfrak{g}_0: [H,X]=\lambda(H)X\;\;\forall H\in\mathfrak{a}_0 \}$$ is the root space. The number $$m_{\lambda}= \text{dim}\,\mathfrak{g}_{\lambda}$$ is called the multiplicity of $$\lambda$$.

Examples
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
 * $$ \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) , $$

\mathbf{A} = \left\{ \begin{pmatrix} r & 0 \\ 0 & r^{-1} \end{pmatrix} \in SL(2,\mathbb{R}) \ | \ r > 0  \right\}, $$

\mathbf{N} = \left\{ \begin{pmatrix} 1 & x \\ 0 & 1 \end{pmatrix} \in SL(2,\mathbb{R}) \ | \  x\in\mathbf{R} \right\}. $$

For the symplectic group G=Sp(2n, R ), a possible Iwasawa decomposition is in terms of


 * $$ \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) , $$

\mathbf{A} = \left\{ \begin{pmatrix} D & 0 \\ 0 & D^{-1} \end{pmatrix} \in Sp(2n,\mathbb{R}) \ | \ D \text{ positive, diagonal} \right\}, $$

\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\}. $$

Non-Archimedean Iwasawa decomposition
There is an analog to the above Iwasawa decomposition for a non-Archimedean field $$F$$: In this case, the group $$GL_n(F)$$ can be written as a product of the subgroup of upper-triangular matrices and the (maximal compact) subgroup $$GL_n(O_F)$$, where $$O_F$$ is the ring of integers of $$F$$.