Lie group decomposition

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The term Lie group decomposition is used for a generic decomposition applicable to one or more Lie groups (in most cases, these are algebraic groups) as a product of smaller Lie groups.

List of decompositions

Name of decomposition Type of Lie group or algebraic group to which it applies Form of decomposition
Bruhat decomposition reductive algebraic group (in particular, semisimple algebraic group) BWB where B is a Borel subgroup and W is a finite group called a Weyl group. The decomposition is a a disjoint union of double cosets BwB, w \in W.
Iwasawa decomposition semisimple Lie group KAN where K is a maximal compact subgroup, A is an abelian subgroup (analogous to diagonal matrices) and N is a maximal unipotent subgroup.
Langlands decomposition parabolic subgroup of a Lie group MAN where M is semisimple, A is an abelian subgroup (analogous to diagonal matrices) and N is a maximal unipotent subgroup.
polar decomposition semisimple Lie group KAK where K is a maximal compact subgroup and A is an abelian subgroup.