# 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 |

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Iwasawa decomposition.

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