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  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 , ).
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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.
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Iwasawa decomposition. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Iwasawa_decomposition&oldid=21877