Lie Groups pp 197-204 | Cite as

The Iwasawa Decomposition

  • Daniel Bump
Part of the Graduate Texts in Mathematics book series (GTM, volume 225)


Let us begin this topic with an example. Let G = GL(n, ℂ). It is the complexification of K= U(n), which is a maximal compact subgroup. Let T be the maximal torus of K consisting of diagonal matrices whose eigenvalues have absolute value 1. The complexification T of T can be factored as TA, where A is the group of diagonal matrices whose eigenvalues are positive real numbers. Let B be the group of upper triangular matrices in G,and let B 0 be the subgroup of elements of B whose diagonal entries are positive real numbers. Finally, let N be the subgroup of unipotent elements of B. Recalling that a matrix is called unipotent if its only eigenvalue is 1, the elements of N are upper triangular matrices whose diagonal entries are all equal to 1. We may factor B = TN and B 0 = AN. The subgroup N is normal in B and B 0, so these decompositions are semidirect products.


Maximal Torus Triangular Matrice Borel Subgroup Maximal Compact Subgroup Unipotent Element 
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Copyright information

© Springer Science+Business Media New York 2004

Authors and Affiliations

  • Daniel Bump
    • 1
  1. 1.Department of MathematicsStanford UniversityStanfordUSA

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