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=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

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

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

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

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 $F$.

See also

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

References

• Fedenko, A.S.; Shtern, A.I. (2001) [1994], "Iwasawa decomposition", Encyclopedia of Mathematics, EMS Press
• Knapp, A. W. (2002). Lie groups beyond an introduction (2nd ed.). ISBN 9780817642594.