Iwasawa decomposition

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In mathematics, the Iwasawa decomposition (aka KAN from its expression) of a semisimple Lie group generalises the way a square real matrix can be written as a product of an orthogonal matrix and an upper triangular matrix (QR decomposition, a consequence of Gram–Schmidt orthogonalization). It is named after Kenkichi Iwasawa, the Japanese mathematician who developed this method.[1]


  • G is a connected semisimple real Lie group.
  • is the Lie algebra of G
  • is the complexification of .
  • θ is a Cartan involution of
  • is the corresponding Cartan decomposition
  • is a maximal abelian subalgebra of
  • Σ is the set of restricted roots of , corresponding to eigenvalues of acting on .
  • Σ+ is a choice of positive roots of Σ
  • is a nilpotent Lie algebra given as the sum of the root spaces of Σ+
  • K, A, N, are the Lie subgroups of G generated by and .

Then the Iwasawa decomposition of is

and the Iwasawa decomposition of G is

meaning there is an analytic diffeomorphism (but not a group homomorphism) from the manifold to the Lie group , sending .

The dimension of A (or equivalently of ) is equal to the real rank of G.

Iwasawa decompositions also hold for some disconnected semisimple groups G, where K becomes a (disconnected) maximal compact subgroup provided the center of G is finite.

The restricted root space decomposition is

where is the centralizer of in and is the root space. The number is called the multiplicity of .


If G=SLn(R), then we can take K to be the orthogonal matrices, A to be the positive diagonal matrices with determinant 1, and N to be the unipotent group consisting of upper triangular matrices with 1s on the diagonal.

For the case of n=2, Iwasawa decomposition of G=SL(2,R) is in terms of

For the symplectic group G=Sp(2n, R ), a possible Iwasawa-decomposition is in terms of

Non-Archimedean Iwasawa decomposition[edit]

There is an analog to the above Iwasawa decomposition for a non-Archimedean field : In this case, the group can be written as a product of the subgroup of upper-triangular matrices and the (maximal compact) subgroup , where is the ring of integers of .[2]

See also[edit]


  1. ^ Iwasawa, Kenkichi (1949). "On Some Types of Topological Groups". Annals of Mathematics. 50 (3): 507–558. doi:10.2307/1969548. JSTOR 1969548.
  2. ^ Bump, Daniel (1997), Automorphic forms and representations, Cambridge: Cambridge University Press, doi:10.1017/CBO9780511609572, ISBN 0-521-55098-X, Prop. 4.5.2