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

Then the Iwasawa decomposition of ${𝔤}_0$ is

${𝔤}_0={𝔨}_0\oplus {𝔞}_0\oplus {𝔫}_0$

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 ${𝔞}_0$) 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

${𝔤}_0={𝔪}_0\oplus {𝔞}_0{\oplus }_{\lambda \in \mathrm{\Sigma }}{𝔤}_{\lambda }$

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

Examples

If G=SL_n(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

$𝐊=\{\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right)\in SL(2,ℝ)|\theta \in 𝐑\}\cong SO(2),$

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

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

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

$𝐊=Sp(2n,ℝ)\cap SO(2n)=\{\left(\begin{array}{cc}A& B\\ -B& A\end{array}\right)\in Sp(2n,ℝ)|A+iB\in U(n)\}\cong U(n),$

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

$𝐍=\{\left(\begin{array}{cc}N& M\\ 0& N^{-T}\end{array}\right)\in Sp(2n,ℝ)|N\text{ upper triangular with diagonal elements = 1},N{M}^T=M{N}^T\}.$

Non-Archimedean Iwasawa decomposition

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

See also

• Lie group decompositions
• Root system of a semi-simple Lie algebra

References

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