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.
is the Lie algebra of G
is the complexification of
. - θ is a Cartan involution of
![{\displaystyle {\mathfrak {g}}_{0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/638ad4f9e9b8e2076287c479775d87ac30395794)
is the corresponding Cartan decomposition
is a maximal abelian subalgebra of ![{\displaystyle {\mathfrak {p}}_{0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/71c07b225760ae2da93b58581b8da3105b42b19c)
- Σ is the set of restricted roots of
, corresponding to eigenvalues of
acting on
. - Σ+ is a choice of positive roots of Σ
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
and
.
Then the Iwasawa decomposition of
is
![{\displaystyle {\mathfrak {g}}_{0}={\mathfrak {k}}_{0}\oplus {\mathfrak {a}}_{0}\oplus {\mathfrak {n}}_{0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/412f9b560145cabb899517d334e5118191be6713)
and the Iwasawa decomposition of G is
![{\displaystyle G=KAN}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd843cd658e6341a452a000cdffe583a789c56a6)
meaning there is an analytic diffeomorphism (but not a group homomorphism) from the manifold
to the Lie group
, sending
.
The dimension of A (or equivalently of
) 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 }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a565fd8781c52beeb4513e868577d556ad4bfd70)
where
is the centralizer of
in
and
is the root space. The number
is called the multiplicity of
.
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
![{\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),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91a5e54ac0e532083ae6c1876a70e6551d959891)
![{\displaystyle \mathbf {A} =\left\{{\begin{pmatrix}r&0\\0&r^{-1}\end{pmatrix}}\in SL(2,\mathbb {R} )\ |\ r>0\right\},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b0a643f272d3ab5a4ba324fd52ca903e1f55bcb1)
![{\displaystyle \mathbf {N} =\left\{{\begin{pmatrix}1&x\\0&1\end{pmatrix}}\in SL(2,\mathbb {R} )\ |\ x\in \mathbf {R} \right\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/006c1abced207a7044bef7adb12e13796849da62)
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),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31c545c26f2084b36f7f54c8a09726b7ee4a2530)
![{\displaystyle \mathbf {A} =\left\{{\begin{pmatrix}D&0\\0&D^{-1}\end{pmatrix}}\in Sp(2n,\mathbb {R} )\ |\ D{\text{ positive, diagonal}}\right\},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3160ba6bb8c6c37de3275aa70b6007c58bd26e91)
![{\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\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff3a1342f2ecc85d42d8bb73c23e94d7692c4124)
Non-Archimedean Iwasawa decomposition
There is an analog to the above Iwasawa decomposition for a non-Archimedean field
: In this case, the group
can be written as a product of the subgroup of upper-triangular matrices and the (maximal compact) subgroup
, where
is the ring of integers of
.[2]
See also
References
- ^ Iwasawa, Kenkichi (1949). "On Some Types of Topological Groups". Annals of Mathematics. 50 (3): 507–558. doi:10.2307/1969548. JSTOR 1969548.
- ^ Bump, Daniel (1997), Automorphic forms and representations, Cambridge: Cambridge University Press, doi:10.1017/CBO9780511609572, ISBN 0-521-55098-X, Prop. 4.5.2