An Iwasawa decomposition of a connected real semisimple Lie group 
with finite group center is a factorization
 |
where 
is a maximal compact subgroup, 
is a subgroup that is an abelian
group, and 
is a subgroup that is a nilpotent
group. Multiplication gives an analytic diffeomorphism from 
onto 
, so each element 
has a unique factorization 
, where 
, 
, and 
(Iwasawa 1949, Helgason 1978).
For example, the Iwasawa decomposition of 
has
 |
and 
equal to the upper triangular matrices
with positive diagonal entries and determinant 1.
It is the group-theoretic analog of the QR decomposition.
For 
considered as a real Lie group, one has 
, and the decomposition underlies the Hermitian
matrix model 
of hyperbolic space. Versions for loop
groups are a central step in the DPW method.
