Iwasawa decomposition

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.^[1]

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=SL_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-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 $F$. ^[2]

See also

• Lie group decompositions

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. Bump, Automorphic Forms and Representations, Prop. 4.5.2

• 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)