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 (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*=*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*, 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

**^**Iwasawa, Kenkichi (1949). "On Some Types of Topological Groups".*Annals of Mathematics*.**50**(3): 507–558. doi:10.2307/1969548. JSTOR 1969548.**^**Bump, Daniel (1997),*Automorphic forms and representations*, Cambridge: Cambridge University Press, doi:10.1017/CBO9780511609572, ISBN 0-521-55098-X, Prop. 4.5.2

- 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.