Iwasawa decomposition

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In mathematics, the Iwasawa decomposition KAN 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 (a consequence of Gram–Schmidt orthogonalization). It is named after Kenkichi Iwasawa, the Japanese mathematician who developed this method.[1]

Definition[edit]

  • 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

The dimension of A (or equivalently of ) is called 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 .

Examples[edit]

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

Non-Archimedean Iwasawa decomposition[edit]

There is an analogon 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]

References[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, Automorphic Forms and Representations, Prop. 4.5.2