Differential of the first kind

In mathematics, the Iwasawa decomposition KAN 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 (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

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

and the Iwasawa decomposition of G is

$G=KAN$

The dimension of A (or equivalently of ${\mathfrak {a}}_{0}$) is called 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=GL_n(R), then we can take K to be the orthogonal matrices, A to be the positive diagonal matrices, and N to be the unipotent group consisting of upper triangular matrices with 1s on the diagonal.

Non-archimedian Iwasawa decomposition

There is an analogon 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

References

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• A. W. Knapp, Structure theory of semisimple Lie groups, in ISBN 0-8218-0609-2: Representation Theory and Automorphic Forms: Instructional Conference, International Centre for Mathematical Sciences, March 1996, Edinburgh, Scotland (Proceedings of Symposia in Pure Mathematics) by T. N. Bailey (Editor), Anthony W. Knapp (Editor)

• Iwasawa, Kenkichi: On some types of topological groups. Annals of Mathematics (2) 50, (1949), 507–558.

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