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]

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

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

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

and the Iwasawa decomposition of G is

${\displaystyle G=KAN}${\displaystyle G=KAN}

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

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

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

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

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

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

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

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

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

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

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

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

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

• 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.). Springer. ISBN 9780817642594.

• Lie groups

• Articles with short description
• Short description matches Wikidata