Math 2.0

Topological groups, Lie groups Structure of Lie groups Theorem

Iwasawa decomposition of a Lie group

Builds on Semisimple Lie groupCartan decomposition of a Lie groupNilpotent Lie groupExponential map of a Lie group

Statement

Let GG be a connected semisimple Lie group with finite center and Lie algebra 𝔤\mathfrak{g}. Let 𝔤=𝔨⊕𝔭\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p} be the Cartan decomposition for a Cartan involution, as in the Cartan decomposition. Let KK be the connected subgroup with Lie algebra 𝔨\mathfrak{k}. Let 𝔞⊆𝔭\mathfrak{a} \subseteq \mathfrak{p} be a maximal abelian subspace. For a linear form λ≠0\lambda \ne 0 on 𝔞\mathfrak{a}, put 𝔤λ={X∈𝔤:[H,X]=λ(H)X for all H∈𝔞}\mathfrak{g}_\lambda = \set{X \in \mathfrak{g} : [H, X] = \lambda(H) X \text{ for all } H \in \mathfrak{a}}, and let Σ\Sigma be the set of λ\lambda with 𝔤λ≠0\mathfrak{g}_\lambda \ne 0, the restricted roots. Fix H0∈𝔞H_0 \in \mathfrak{a} with λ(H0)≠0\lambda(H_0) \ne 0 for all λ∈Σ\lambda \in \Sigma, put Σ+={λ∈Σ:λ(H0)>0}\Sigma^+ = \set{\lambda \in \Sigma : \lambda(H_0) > 0} and 𝔫=⨁λ∈Σ+𝔤λ\mathfrak{n} = \bigoplus_{\lambda \in \Sigma^+} \mathfrak{g}_\lambda. Let AA and NN be the connected subgroups of GG with Lie algebras 𝔞\mathfrak{a} and 𝔫\mathfrak{n}. Then AA is abelian, NN is nilpotent, both are closed and simply connected, and the map K×A×N→G,(k,a,n)↦kan,K \times A \times N \to G, \quad (k, a, n) \mapsto kan, is a diffeomorphism.

Remarks

On the Lie algebra level, 𝔤=𝔨⊕𝔞⊕𝔫\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{a} \oplus \mathfrak{n} as vector spaces; this follows from the restricted root space decomposition and the fact that θ\theta maps 𝔤λ\mathfrak{g}_\lambda to 𝔤−λ\mathfrak{g}_{-\lambda}. One then shows that ANAN is a closed simply connected solvable subgroup, and that K×AN→GK \times AN \to G is a bijective local diffeomorphism.

Example: G=SLn(ℝ)G = \mathrm{SL}_n(\R) with K=SO(n)K = \mathrm{SO}(n). Here AA is the group of diagonal matrices with positive entries and determinant 11, and NN is the group of upper triangular matrices with 11 on the diagonal. The decomposition g=kang = kan is the Gram–Schmidt process applied to the columns of gg, also known as the QR decomposition.

Going deeper

For a student

The Iwasawa decomposition shows that G/KG/K is diffeomorphic to the solvable group ANAN. So the symmetric space G/KG/K is a solvable Lie group with a left-invariant metric. The subgroup MANMAN, with MM the centralizer of 𝔞\mathfrak{a} in KK, is the minimal parabolic subgroup. Inducing to GG the representations of MANMAN that are trivial on NN gives the principal series representations. The dimension of 𝔞\mathfrak{a} is the real rank of GG.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Exponential map of a Lie group, concept, no evidence levelExponential map of a …conceptMaximal compact subgroup, concept, no evidence levelMaximal compact subgr…conceptNilpotent Lie group, concept, no evidence levelNilpotent Lie groupconceptSemisimple Lie group, concept, no evidence levelSemisimple Lie groupconceptCartan decomposition of a Lie group, statement, no evidence levelCartan decomposition …statementIwasawa decomposition of a Lie group, statement, no evidence levelIwasawa decomposition…statement
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  1. Exponential map of a Lie group concept
  2. Maximal compact subgroup concept
  3. Nilpotent Lie group concept
  4. Semisimple Lie group concept
  5. Cartan decomposition of a Lie group statement
  6. Iwasawa decomposition of a Lie group statement

Where this leads

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Further reading

  1. Web https://en.wikipedia.org/wiki/Iwasawa_decomposition
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Kindstatement, theorem
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MSC22E46, 22E15
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AboutSemisimple Lie group
UsesSemisimple Lie group, Cartan decomposition of a Lie group, Nilpotent Lie group, Exponential map of a Lie group
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Changed2 November 2026

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