nLab Iwasawa decomposition

Idea

Am Iwasawa decomposition is an analytic diffeomorphism $G \,\simeq\, K \times A \times N$ between

1. a connected real semisimple Lie group $G$

2. a product manifold $K \times A \times N$,

   where $K$, $A$ and $N$ are

   • analytic$\;$subgroups of $G$,

   • with $A$ and $N$ simply connected,

   • integratingLie algebras $k,a,n$, respectively, where

   • $k\oplus p$ is the Cartan decomposition? of the Lie algebra of $G$,

   • $a$ is the maximal Abelian subalgebra of $p$

   • $n$ is a certain nilpotent Lie subalgebra of $p$ with $p \simeq a\oplus n$ as vector spaces.

The Iwasawa decomposition generalizes a statement expressing the QR decomposition of real matrices.

References

Named after:

• Kenkichi Iwasawa, On some types of topological groups, Ann. Math. 50 2 (1949) 507558 [doi:10.2307/1969548, jstor:]

See also:

• Daniel Bump, The Iwasawa decomposition, in: Lie Groups, Graduate Texts in Mathematics 225, Springer (2004) [doi:10.1007/978-1-4757-4094-3_29]

• eom, Iwasawa decomposition

• Wikipedia, Iwasawa decomposition

There is also a quantum group version, discussed e.g. in

• Branislav Jurčo, P. Šťovíček, Quantum dressing orbits on compact groups, Commun. Math. Physics 152 (1993) 97–126