Lie Group Decomposition - List of Decompositions

List of Decompositions

• The Bruhat decomposition G = BWB of a semisimple algebraic group into double cosets of a Borel subgroup can be regarded as a general expression of the principle of Gauss–Jordan elimination, which generically writes a matrix as the product of an upper triangular matrix with a lower triangular matrix—but with exceptional cases. It is related to the Schubert cell decomposition of Grassmannians: see Weyl group for more details.
• The Cartan decomposition writes a semisimple real Lie algebra as the sum of eigenspaces of a Cartan involution.
• The Iwasawa decomposition G = KAN of a semisimple group G as the product of compact, abelian, and nilpotent subgroups generalises the way a square real matrix can be written as a product of an orthogonal matrix and an upper triangular matrix (a consequence of Gram–Schmidt orthogonalization).
• The Langlands decomposition P = MAN writes a parabolic subgroup P of a Lie group as the product of semisimple, abelian, and nilpotent subgroups.
• The Levi decomposition writes a finite dimensional Lie algebra as a semidirect product of a normal solvable subalgebra by a semisimple subalgebra.
• The Polar decomposition G = KAK writes a semisimple Lie group G in terms of a maximal compact subgroup K and an abelian subgroup A.