Iwasawa decomposition
E542127
The Iwasawa decomposition is a fundamental factorization in Lie group theory that expresses a semisimple Lie group as a product of a maximal compact subgroup, a maximal abelian subgroup, and a nilpotent subgroup, playing a key role in representation theory and harmonic analysis.
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
decomposition in Lie theory
ⓘ
mathematical concept ⓘ |
| alsoKnownAs | KAN decomposition NERFINISHED ⓘ |
| appliesTo |
real semisimple Lie group
ⓘ
reductive Lie group ⓘ semisimple Lie group ⓘ |
| component |
maximal abelian subgroup A
ⓘ
maximal compact subgroup K ⓘ nilpotent subgroup N ⓘ |
| componentRole |
A corresponds to split part of Cartan subgroup
ⓘ
K captures compact directions of G ⓘ N corresponds to sum of positive root spaces ⓘ |
| context |
real forms of complex semisimple Lie groups
ⓘ
structure theory of Lie groups ⓘ |
| ensures | G is diffeomorphic to K × A × N as manifolds ⓘ |
| expresses | Lie group as product of subgroups ⓘ |
| field |
Lie group theory
ⓘ
differential geometry ⓘ harmonic analysis ⓘ representation theory ⓘ |
| generalizes | classical polar decomposition of matrices ⓘ |
| hasForm | G = K A N ⓘ |
| implies | every element g in G can be written uniquely as k a n ⓘ |
| namedAfter | Kenkichi Iwasawa NERFINISHED ⓘ |
| property |
A is abelian and consists of semisimple elements
ⓘ
K is a maximal compact subgroup of G ⓘ N is connected and nilpotent ⓘ multiplication map K × A × N → G is a diffeomorphism ⓘ |
| relatedTo |
Bruhat decomposition
NERFINISHED
ⓘ
Cartan decomposition NERFINISHED ⓘ Langlands decomposition NERFINISHED ⓘ polar decomposition ⓘ |
| requires |
choice of maximal abelian subspace of a Cartan subalgebra
ⓘ
choice of maximal compact subgroup K ⓘ choice of positive root system ⓘ |
| specialCase | G = SL(2,R) with K = SO(2), A diagonal, N unipotent upper triangular ⓘ |
| usedFor |
Fourier analysis on non-compact Lie groups
ⓘ
construction of invariant measures ⓘ parametrization of symmetric spaces ⓘ |
| usedIn |
Plancherel formula for semisimple Lie groups
NERFINISHED
ⓘ
analysis of eigenfunctions of invariant differential operators ⓘ construction of principal series representations ⓘ description of Haar measure on G in KAN coordinates ⓘ harmonic analysis on semisimple Lie groups ⓘ representation theory of real reductive groups ⓘ spherical functions on Lie groups ⓘ study of non-compact Riemannian symmetric spaces ⓘ theory of automorphic forms ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.