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]
Definition
- 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
.
Examples
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, the 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 decomposition
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 also
References
- Fedenko, A.S.; Shtern, A.I. (2001) [1994], "Iwasawa decomposition", Encyclopedia of Mathematics, EMS Press
- Knapp, A. W. (2002). Lie groups beyond an introduction (2nd ed.). ISBN 9780817642594.