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.

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 subspace 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 $mathfrak\{g\}\_0$:$mathfrak\{g\}\_0\; =\; mathfrak\{k\}\_0\; +\; mathfrak\{a\}\_0\; +\; mathfrak\{n\}\_0$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.

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.

ee also

*Lie group decompositions

References

*springer|id=I/i053060|first1=A.S. |last1=Fedenko|first2=A.I.|last2= Shtern
*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.