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 Japan ese 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

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

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

Examples

If G=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

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

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

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

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

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

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

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

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

• Hazewinkel, Michiel, ed. (2001), "Iwasawa decomposition", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Iwasawa_decomposition&oldid=21877 
• Knapp, A. W. (2002). Lie groups beyond an introduction (2nd ed.). ISBN 9780817642594.