Iwasawa decomposition
The unique representation of an arbitrary element 
of a non-compact connected semi-simple real Lie group 
as a product 
of elements 
of analytic subgroups 
, respectively, where 
, 
and 
are defined as follows. Let 
be a Cartan decomposition of the Lie algebra 
of 
; let 
be the maximal commutative subspace of the space 
, and let 
be a nilpotent Lie subalgebra of 
such that 
is the linear hull of the root vectors in some system of positive roots with respect to 
. The decomposition of the Lie algebra as the direct sum of the subalgebras 
, 
and 
is called the Iwasawa decomposition [1] of the semi-simple real Lie algebra 
. The groups 
, 
and 
are defined to be the analytic subgroups of 
corresponding to the subalgebras 
, 
and 
, respectively. The groups 
, 
and 
are closed; 
and 
are simply-connected; 
contains the centre of 
, and the image of 
under the adjoint representation of 
is a maximal compact subgroup of the adjoint group of 
. The mapping 
is an analytic diffeomorphism of the manifold 
onto the Lie group 
. The Iwasawa decomposition plays a fundamental part in the representation theory of semi-simple Lie groups. The Iwasawa decomposition can be defined also for connected semi-simple algebraic groups over a 
-adic field (or, more generally, for groups of 
-adic type) (see [4], [5]).
References
| [1] | K. Iwasawa, "On some types of topological groups" Ann. of Math. , 50 (1949) pp. 507–558 MR0029911 Zbl 0034.01803 |
| [2] | M.A. Naimark, "Theory of group representations" , Springer (1982) (Translated from Russian) MR0793377 Zbl 0484.22018 |
| [3] | S. Helgason, "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press (1978) MR0514561 Zbl 0451.53038 |
| [4] | F. Bruhat, "Sur une classe de sous-groupes compacts maximaux des groupes de Chevalley sur un corps  -adique" Publ. Math. IHES , 23 (1964) pp. 45–74 MR179298 |
| [5] | N. Iwahori, H. Matsumoto, "On some Bruhat decomposition and the structure of the Hecke rings of  -adic Chevalley groups" Publ. Math. IHES , 25 (1965) pp. 5–48 MR185016 |
Comments
An example of an Iwasawa decomposition is 
with 
, 
the subgroup of diagonal matrices of 
and 
a lower triangular matrix with 
's on the diagonal. So, in particular, every element of 
gets written as a product of a special orthogonal matrix and a lower triangular matrix.
References
| [a1] | S. Helgason, "Groups and geometric analysis" , Acad. Press (1984) pp. Chapt. II, Sect. 4 MR0754767 Zbl 0543.58001 |
Iwasawa decomposition. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Iwasawa_decomposition&oldid=14077