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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 (a consequence of Gram–Schmidt orthogonalization). It is named after Kenkichi Iwasawa, the Japanese mathematician who developed this method.

Contents

Definition

  • G is a connected semisimple real Lie group.
  • g 0 is the Lie algebra of G
  • g is the complexification of g 0 .
  • θ is a Cartan involution of g 0
  • g 0 = k 0 ⊕ p 0 is the corresponding Cartan decomposition
  • a 0 is a maximal abelian subalgebra of p 0
  • Σ is the set of restricted roots of a 0 , corresponding to eigenvalues of a 0 acting on g 0 .
  • Σ+ is a choice of positive roots of Σ
  • n 0 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 k 0 , a 0 and n 0 .

    • Then the Iwasawa decomposition of g 0 is

    g 0 = k 0 ⊕ a 0 ⊕ n 0

    and the Iwasawa decomposition of G is

    G = K A N

    The dimension of A (or equivalently of a 0 ) 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

    g 0 = m 0 ⊕ a 0 ⊕ λ ∈ Σ g λ

    where m 0 is the centralizer of a 0 in k 0 and g λ = { X ∈ g 0 : [ H , X ] = λ ( H ) X ∀ H ∈ a 0 } is the root space. The number m λ = dim g λ is called the multiplicity of λ .

    Examples

    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

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

    References

    Iwasawa decomposition Wikipedia
    (Text) CC BY-SA


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