Iwasawa decomposition

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

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

g 0 = k 0 ⊕ a 0 ⊕ n 0 {\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

G = K A N {\displaystyle G=KAN} {\displaystyle G=KAN}

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

The dimension of A (or equivalently of a 0 {\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

g 0 = m 0 ⊕ a 0 ⊕ λ ∈ Σ g λ {\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 m 0 {\displaystyle {\mathfrak {m}}_{0}} {\displaystyle {\mathfrak {m}}_{0}} is the centralizer of a 0 {\displaystyle {\mathfrak {a}}_{0}} {\displaystyle {\mathfrak {a}}_{0}} in k 0 {\displaystyle {\mathfrak {k}}_{0}} {\displaystyle {\mathfrak {k}}_{0}} and g λ = { X ∈ g 0 : [ H , X ] = λ ( H ) X ∀ H ∈ a 0 } {\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 m λ = dim g λ {\displaystyle m_{\lambda }={\text{dim}}\,{\mathfrak {g}}_{\lambda }} {\displaystyle m_{\lambda }={\text{dim}}\,{\mathfrak {g}}_{\lambda }} is called the multiplicity of λ {\displaystyle \lambda } {\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, the Iwasawa decomposition of G = SL(2, R) is in terms of

K = { ( cos ⁡ θ − sin ⁡ θ sin ⁡ θ cos ⁡ θ ) ∈ S L ( 2 , R )   |   θ ∈ R } ≅ S O ( 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 {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),}
A = { ( r 0 0 r − 1 ) ∈ S L ( 2 , R )   |   r > 0 } , {\displaystyle \mathbf {A} =\left\{{\begin{pmatrix}r&0\\0&r^{-1}\end{pmatrix}}\in SL(2,\mathbb {R} )\ |\ r>0\right\},} {\displaystyle \mathbf {A} =\left\{{\begin{pmatrix}r&0\\0&r^{-1}\end{pmatrix}}\in SL(2,\mathbb {R} )\ |\ r>0\right\},}
N = { ( 1 x 0 1 ) ∈ S L ( 2 , R )   |   x ∈ R } . {\displaystyle \mathbf {N} =\left\{{\begin{pmatrix}1&x\\0&1\end{pmatrix}}\in SL(2,\mathbb {R} )\ |\ x\in \mathbf {R} \right\}.} {\displaystyle \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

K = S p ( 2 n , R ) ∩ S O ( 2 n ) = { ( A B − B A ) ∈ S p ( 2 n , R )   |   A + i B ∈ U ( n ) } ≅ 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 {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),}
A = { ( D 0 0 D − 1 ) ∈ S p ( 2 n , R )   |   D  positive, diagonal } , {\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 {A} =\left\{{\begin{pmatrix}D&0\\0&D^{-1}\end{pmatrix}}\in Sp(2n,\mathbb {R} )\ |\ D{\text{ positive, diagonal}}\right\},}
N = { ( N M 0 N − T ) ∈ S p ( 2 n , R )   |   N  upper triangular with diagonal elements = 1 ,   N M T = M N T } . {\displaystyle \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\}.} {\displaystyle \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\}.}

Obtaining the matrices appearing in the decomposition above can be reduced to the calculation of matrix square roots, matrix inverses and performing a QR decomposition.[2]

Non-Archimedean Iwasawa decomposition

There is an analog to the above Iwasawa decomposition for a non-Archimedean field F {\displaystyle F} {\displaystyle F}: In this case, the group G L n ( F ) {\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 G L n ( O F ) {\displaystyle GL_{n}(O_{F})} {\displaystyle GL_{n}(O_{F})}, where O F {\displaystyle O_{F}} {\displaystyle O_{F}} is the ring of integers of F {\displaystyle F} {\displaystyle F}.[3]

See also

References

  1. Iwasawa, Kenkichi (1949). "On Some Types of Topological Groups". Annals of Mathematics. 50 (3): 507–558. doi:10.2307/1969548. JSTOR 1969548.
  2. Houde, Martin; McCutcheon, Will; Quesada, Nicolas (2024). "Matrix decompositions in quantum optics: Takagi/Autonne, Bloch–Messiah/Euler, Iwasawa, and Williamson". Canadian Journal of Physics. 102 (10): 497–597. doi:10.1139/cjp-2024-0070. hdl:1807/139527.
  3. Bump, Daniel (1997), Automorphic forms and representations, Cambridge: Cambridge University Press, doi:10.1017/CBO9780511609572, ISBN 0-521-55098-X, Prop. 4.5.2