IWASAWA DECOMPOSITION

The 'Iwasawa decomposition' KAN of a semisimple 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).

Contents

Definition

Examples

See also

External link

References

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
:$G=KAN$
Iwasawa decompositions also hold for some disconnected semisimple groups ''G'', where ''K'' becomes a (disconnected) maximal compact subgroup.

Examples

If ''G''=''GL_n''('R'), then we can take ''K'' to be the orthogonal matrices, ''A'' to be the diagonal matrices, and ''N'' to be the unipotent matrices (upper triangular matrices with 1s on the diagonal).

See also

Lie group decompositions

External link

★

References

★ 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.