Iwasawa decomposition

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]

DefinitionEdit

  • 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  .

ExamplesEdit

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 decompositionEdit

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 alsoEdit

ReferencesEdit

  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