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

${\mathfrak{g}}_0={\mathfrak{m}}_0\oplus {\mathfrak{a}}_0{\oplus }_{\lambda \in \mathrm{\Sigma }}{\mathfrak{g}}_{\lambda }$

where ${\mathfrak{m}}_0$ is the centralizer of ${\mathfrak{a}}_0$ in ${\mathfrak{k}}_0$ and ${\mathfrak{g}}_{\lambda }=\{X\in {\mathfrak{g}}_0:[H,X]=\lambda (H)X\mathrm{\forall }H\in {\mathfrak{a}}_0\}$ 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 $G{L}_n(F)$ can be written as a product of the subgroup of upper-triangular matrices and the (maximal compact) subgroup $G{L}_n(O_F)$, 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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