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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

$$\mathfrak{g}_0 = \mathfrak{k}_0 \oplus \mathfrak{a}_0 \oplus \mathfrak{n}_0$$

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=SLn(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-Archimedean 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. [1]

Lie group decompositions

References

Fedenko, A.S.; Shtern, A.I. (2001), "I/i053060", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
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.

Bump, Automorphic Forms and Representations, Prop. 4.5.2

Mathematics Encyclopedia