Iwasawa decomposition

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).

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 links

• A.I. Shtern, A.S. Fedenko (2001) [1994], "Iwasawa decomposition", Encyclopedia of Mathematics, EMS Press

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.