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 Japan ese mathematician who developed this method.

Definition

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

Then the Iwasawa decomposition of [math]\displaystyle{ \mathfrak{g}_0 }[/math] is

[math]\displaystyle{ \mathfrak{g}_0 = \mathfrak{k}_0 \oplus \mathfrak{a}_0 \oplus \mathfrak{n}_0 }[/math]

and the Iwasawa decomposition of G is

[math]\displaystyle{ G=KAN }[/math]

meaning there is an analytic diffeomorphism (but not a group homomorphism) from the manifold [math]\displaystyle{ K \times A \times N }[/math] to the Lie group [math]\displaystyle{ G }[/math], sending [math]\displaystyle{ (k,a,n) \mapsto kan }[/math].

The dimension of A (or equivalently of [math]\displaystyle{ \mathfrak{a}_0 }[/math]) 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

[math]\displaystyle{ \mathfrak{g}_0 = \mathfrak{m}_0\oplus\mathfrak{a}_0\oplus_{\lambda\in\Sigma}\mathfrak{g}_{\lambda} }[/math]

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

Examples

If G=SL_n(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, the Iwasawa decomposition of G=SL(2,R) is in terms of

[math]\displaystyle{ \mathbf{K} = \left\{ \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix} \in SL(2,\mathbb{R}) \ | \ \theta\in\mathbf{R} \right\} \cong SO(2) , }[/math]

[math]\displaystyle{ \mathbf{A} = \left\{ \begin{pmatrix} r & 0 \\ 0 & r^{-1} \end{pmatrix} \in SL(2,\mathbb{R}) \ | \ r \gt 0 \right\}, }[/math]

[math]\displaystyle{ \mathbf{N} = \left\{ \begin{pmatrix} 1 & x \\ 0 & 1 \end{pmatrix} \in SL(2,\mathbb{R}) \ | \ x\in\mathbf{R} \right\}. }[/math]

For the symplectic group G=Sp(2n, R ), a possible Iwasawa decomposition is in terms of

[math]\displaystyle{ \mathbf{K} = Sp(2n,\mathbb{R})\cap SO(2n) = \left\{ \begin{pmatrix} A & B \\ -B & A \end{pmatrix} \in Sp(2n,\mathbb{R}) \ | \ A+iB \in U(n) \right\} \cong U(n) , }[/math]

[math]\displaystyle{ \mathbf{A} = \left\{ \begin{pmatrix} D & 0 \\ 0 & D^{-1} \end{pmatrix} \in Sp(2n,\mathbb{R}) \ | \ D \text{ positive, diagonal} \right\}, }[/math]

[math]\displaystyle{ \mathbf{N} = \left\{ \begin{pmatrix} N & M \\ 0 & N^{-T} \end{pmatrix} \in Sp(2n,\mathbb{R}) \ | \ N \text{ upper triangular with diagonal elements = 1},\ NM^T = MN^T \right\}. }[/math]

Non-Archimedean Iwasawa decomposition

There is an analog to the above Iwasawa decomposition for a non-Archimedean field [math]\displaystyle{ F }[/math]: In this case, the group [math]\displaystyle{ GL_n(F) }[/math] can be written as a product of the subgroup of upper-triangular matrices and the (maximal compact) subgroup [math]\displaystyle{ GL_n(O_F) }[/math], where [math]\displaystyle{ O_F }[/math] is the ring of integers of [math]\displaystyle{ F }[/math].

See also

• Lie group decompositions
• Root system of a semi-simple Lie algebra

References

• Hazewinkel, Michiel, ed. (2001), "Iwasawa decomposition", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Iwasawa_decomposition&oldid=21877 
• Knapp, A. W. (2002). Lie groups beyond an introduction (2nd ed.). ISBN 9780817642594.