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.[1]

Definition

• G is a connected semisimple real Lie group.
• ${\displaystyle {\mathfrak {g}}_{0}}$ is the Lie algebra of G
• ${\displaystyle {\mathfrak {g}}}$ is the complexification of ${\displaystyle {\mathfrak {g}}_{0}}$.
• θ is a Cartan involution of ${\displaystyle {\mathfrak {g}}_{0}}$
• ${\displaystyle {\mathfrak {g}}_{0}={\mathfrak {k}}_{0}\oplus {\mathfrak {p}}_{0}}$ is the corresponding Cartan decomposition
• ${\displaystyle {\mathfrak {a}}_{0}}$ is a maximal abelian subalgebra of ${\displaystyle {\mathfrak {p}}_{0}}$
• Σ is the set of restricted roots of ${\displaystyle {\mathfrak {a}}_{0}}$, corresponding to eigenvalues of ${\displaystyle {\mathfrak {a}}_{0}}$ acting on ${\displaystyle {\mathfrak {g}}_{0}}$.
• Σ+ is a choice of positive roots of Σ
• ${\displaystyle {\mathfrak {n}}_{0}}$ 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 ${\displaystyle {\mathfrak {k}}_{0},{\mathfrak {a}}_{0}}$ and ${\displaystyle {\mathfrak {n}}_{0}}$.

Then the Iwasawa decomposition of ${\displaystyle {\mathfrak {g}}_{0}}$ is

${\displaystyle {\mathfrak {g}}_{0}={\mathfrak {k}}_{0}\oplus {\mathfrak {a}}_{0}\oplus {\mathfrak {n}}_{0}}$

and the Iwasawa decomposition of G is

${\displaystyle G=KAN}$

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

The dimension of A (or equivalently of ${\displaystyle {\mathfrak {a}}_{0}}$) 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

${\displaystyle {\mathfrak {g}}_{0}={\mathfrak {m}}_{0}\oplus {\mathfrak {a}}_{0}\oplus _{\lambda \in \Sigma }{\mathfrak {g}}_{\lambda }}$

where ${\displaystyle {\mathfrak {m}}_{0}}$ is the centralizer of ${\displaystyle {\mathfrak {a}}_{0}}$ in ${\displaystyle {\mathfrak {k}}_{0}}$ and ${\displaystyle {\mathfrak {g}}_{\lambda }=\{X\in {\mathfrak {g}}_{0}:[H,X]=\lambda (H)X\;\;\forall H\in {\mathfrak {a}}_{0}\}}$ is the root space. The number ${\displaystyle m_{\lambda }={\text{dim}}\,{\mathfrak {g}}_{\lambda }}$ is called the multiplicity of ${\displaystyle \lambda }$.

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, Iwasawa decomposition of G=SL(2,R) is in terms of

${\displaystyle \mathbf {K} =\left\{{\begin{pmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \end{pmatrix}}\in SL(2,\mathbb {R} )\ |\ {\text{ rotation group, angle}}=\theta \right\}\cong SO(2),}$
${\displaystyle \mathbf {A} =\left\{{\begin{pmatrix}r&0\\0&r^{-1}\end{pmatrix}}\in SL(2,\mathbb {R} )\ |\ r>0{\text{ real number, diagonal, }}\det =1\right\},}$
${\displaystyle \mathbf {N} =\left\{{\begin{pmatrix}1&x\\0&1\end{pmatrix}}\in SL(2,\mathbb {R} )\ |\ x\in \mathbf {R} {\text{ upper triangular with diagonals = 1}},\right\}.}$

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

${\displaystyle \mathbf {K} =Sp(2n,\mathbb {R} )\cap SO(2n)=\left\{{\begin{pmatrix}A&B\\-B&A\end{pmatrix}}\in Sp(2n,\mathbb {R} )\ |\ A+iB\in U(n)\right\}\cong U(n),}$
${\displaystyle \mathbf {A} =\left\{{\begin{pmatrix}D&0\\0&D^{-1}\end{pmatrix}}\in Sp(2n,\mathbb {R} )\ |\ D{\text{ positive, diagonal}}\right\},}$
${\displaystyle \mathbf {N} =\left\{{\begin{pmatrix}N&M\\0&N^{-T}\end{pmatrix}}\in Sp(2n,\mathbb {R} )\ |\ N{\text{ upper triangular with diagonals = 1}},\ NM^{T}=MN^{T}\right\}.}$

Non-Archimedean Iwasawa decomposition

There is an analog to the above Iwasawa decomposition for a non-Archimedean field ${\displaystyle F}$: In this case, the group ${\displaystyle GL_{n}(F)}$ can be written as a product of the subgroup of upper-triangular matrices and the (maximal compact) subgroup ${\displaystyle GL_{n}(O_{F})}$, where ${\displaystyle O_{F}}$ is the ring of integers of ${\displaystyle F}$.[2]