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.

Definition

*''G'' is a connected semisimple real Lie group.
*$\mathfrak_0$ is the Lie algebra of ''G''
*$\mathfrak$ is the complexification of $\mathfrak_0$.
*θ is a Cartan involution of $\mathfrak_0$
*$\mathfrak_0 = \mathfrak_0 \oplus \mathfrak_0$ is the corresponding Cartan decomposition
*$\mathfrak_0$ is a maximal abelian subalgebra of $\mathfrak_0$
*Σ is the set of restricted roots of $\mathfrak_0$, corresponding to eigenvalues of $\mathfrak_0$ acting on $\mathfrak_0$.
*Σ⁺ is a choice of positive roots of Σ
*$\mathfrak_0$ 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 $\mathfrak_0, \mathfrak_0$ and $\mathfrak_0$.

Then the Iwasawa decomposition of $\mathfrak_0$ is
:$\mathfrak_0 = \mathfrak_0 \oplus \mathfrak_0 \oplus \mathfrak_0$
and the Iwasawa decomposition of ''G'' is
:$G=KAN$
meaning there is an analytic diffeomorphism (but not a group homomorphism) from the manifold $K \times A \times N$ to the Lie group $G$, sending $(k,a,n) \mapsto kan$.

The dimension of ''A'' (or equivalently of $\mathfrak_0$) 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
:$\mathfrak_0 = \mathfrak_0\oplus\mathfrak_0\oplus_\mathfrak_$
where $\mathfrak_0$ is the centralizer of $\mathfrak_0$ in $\mathfrak_0$ and $\mathfrak_ = \$ is the root space. The number
$m_= \text\,\mathfrak_$ is called the multiplicity of $\lambda$.

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
:$\mathbf = \left\ \cong SO(2) ,$
:$\mathbf = \left\,$
:$\mathbf = \left\.$

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

:$\mathbf = Sp(2n,\mathbb)\cap SO(2n) = \left\ \cong U(n) ,$
:$\mathbf = \left\,$
:$\mathbf = \left\.$

Non-Archimedean Iwasawa decomposition

There is an analog to the above Iwasawa decomposition for a non-Archimedean field $F$: In this case, the group $GL_n(F)$ can be written as a product of the subgroup of upper-triangular matrices and the (maximal compact) subgroup $GL_n(O_F)$, where $O_F$ is the ring of integers of $F$., Prop. 4.5.2

See also

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

References

*
*{{Cite book, title=Lie groups beyond an introduction, authorlink=A. W. Knapp, last=Knapp, first=A. W., ISBN=9780817642594, year=2002, edition=2nd

Lie groups