Iwasawa decomposition

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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.

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[edit] Definition

  • 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 subspace 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

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

and the Iwasawa decomposition of G is

G = KAN

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.

[edit] Examples

If G=GLn(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.

[edit] See also


[edit] References

  • Iwasawa, Kenkichi: On some types of topological groups. Annals of Mathematics (2) 50, (1949), 507–558.
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