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.

Definition[edit]

  • G is a connected semisimple real Lie group.
  •  \mathfrak{g}_0 is the Lie algebra of G
  •  \mathfrak{g} is the complexification of  \mathfrak{g}_0 .
  • θ is a Cartan involution of  \mathfrak{g}_0
  •  \mathfrak{g}_0 = \mathfrak{k}_0 \oplus \mathfrak{p}_0 is the corresponding Cartan decomposition
  •  \mathfrak{a}_0 is a maximal abelian subalgebra of  \mathfrak{p}_0
  • Σ is the set of restricted roots of  \mathfrak{a}_0 , corresponding to eigenvalues of  \mathfrak{a}_0 acting on  \mathfrak{g}_0 .
  • Σ+ is a choice of positive roots of Σ
  •  \mathfrak{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  \mathfrak{k}_0, \mathfrak{a}_0 and  \mathfrak{n}_0 .

Then the Iwasawa decomposition of  \mathfrak{g}_0 is

\mathfrak{g}_0 = \mathfrak{k}_0 \oplus \mathfrak{a}_0 \oplus \mathfrak{n}_0

and the Iwasawa decomposition of G is

G=KAN

The dimension of A (or equivalently of  \mathfrak{a}_0 ) 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

 \mathfrak{g}_0 = \mathfrak{m}_0\oplus\mathfrak{a}_0\oplus_{\lambda\in\Sigma}\mathfrak{g}_{\lambda}

where \mathfrak{m}_0 is the centralizer of \mathfrak{a}_0 in \mathfrak{k}_0 and \mathfrak{g}_{\lambda} = \{X\in\mathfrak{g}_0: [H,X]=\lambda(H)X\;\;\forall H\in\mathfrak{a}_0 \} is the root space. The number m_{\lambda}= \text{dim}\,\mathfrak{g}_{\lambda} is called the multiplicity of \lambda.

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 F: In this case, the group GL_n(F) can be written as a product of the subgroup of upper-triangular matrices and the (maximal compact) subgroup GL_n(O_F), where O_F is the ring of integers of F. [1]

See also[edit]


References[edit]

  • Iwasawa, Kenkichi: On some types of topological groups. Annals of Mathematics (2) 50, (1949), 507–558.
  1. ^ Bump, Automorphic Forms and Representations, Prop. 4.5.2