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F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
G is a connected semisimple real Lie group .
g
0
{\displaystyle {\mathfrak {g}}_{0}}
is the Lie algebra of G
g
{\displaystyle {\mathfrak {g}}}
is the complexification of
g
0
{\displaystyle {\mathfrak {g}}_{0}}
.
θ is a Cartan involution of
g
0
{\displaystyle {\mathfrak {g}}_{0}}
g
0
=
k
0
⊕
p
0
{\displaystyle {\mathfrak {g}}_{0}={\mathfrak {k}}_{0}\oplus {\mathfrak {p}}_{0}}
is the corresponding Cartan decomposition
a
0
{\displaystyle {\mathfrak {a}}_{0}}
is a maximal abelian subalgebra of
p
0
{\displaystyle {\mathfrak {p}}_{0}}
Σ is the set of restricted roots of
a
0
{\displaystyle {\mathfrak {a}}_{0}}
, corresponding to eigenvalues of
a
0
{\displaystyle {\mathfrak {a}}_{0}}
acting on
g
0
{\displaystyle {\mathfrak {g}}_{0}}
.
Σ+ is a choice of positive roots of Σ
n
0
{\displaystyle {\mathfrak {n}}_{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
k
0
,
a
0
{\displaystyle {\mathfrak {k}}_{0},{\mathfrak {a}}_{0}}
and
n
0
{\displaystyle {\mathfrak {n}}_{0}}
.
Then the Iwasawa decomposition of
g
0
{\displaystyle {\mathfrak {g}}_{0}}
is
g
0
=
k
0
⊕
a
0
⊕
n
0
{\displaystyle {\mathfrak {g}}_{0}={\mathfrak {k}}_{0}\oplus {\mathfrak {a}}_{0}\oplus {\mathfrak {n}}_{0}}
and the Iwasawa decomposition of G is
G
=
K
A
N
{\displaystyle G=KAN}
meaning there is an analytic diffeomorphism (but not a group homomorphism) from the manifold
K
×
A
×
N
{\displaystyle K\times A\times N}
to the Lie group
G
{\displaystyle G}
, sending
(
k
,
a
,
n
)
↦
k
a
n
{\displaystyle (k,a,n)\mapsto kan}
.
The dimension of A (or equivalently of
a
0
{\displaystyle {\mathfrak {a}}_{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
g
0
=
m
0
⊕
a
0
⊕
λ
∈
Σ
g
λ
{\displaystyle {\mathfrak {g}}_{0}={\mathfrak {m}}_{0}\oplus {\mathfrak {a}}_{0}\oplus _{\lambda \in \Sigma }{\mathfrak {g}}_{\lambda }}
where
m
0
{\displaystyle {\mathfrak {m}}_{0}}
is the centralizer of
a
0
{\displaystyle {\mathfrak {a}}_{0}}
in
k
0
{\displaystyle {\mathfrak {k}}_{0}}
and
g
λ
=
{
X
∈
g
0
:
[
H
,
X
]
=
λ
(
H
)
X
∀
H
∈
a
0
}
{\displaystyle {\mathfrak {g}}_{\lambda }=\{X\in {\mathfrak {g}}_{0}:[H,X]=\lambda (H )X\;\;\forall H\in {\mathfrak {a}}_{0}\}}
is the root space. The number
m
λ
=
dim
g
λ
{\displaystyle m_{\lambda }={\text{dim}}\,{\mathfrak {g}}_{\lambda }}
is called the multiplicity of
λ
{\displaystyle \lambda }
.
Examples
[ edit ]
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
K
=
{
(
cos
θ
−
sin
θ
sin
θ
cos
θ
)
∈
S
L
(
2
,
R
)
|
θ
∈
R
}
≅
S
O
(
2
)
,
{\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 ),}
A
=
{
(
r
0
0
r
−
1
)
∈
S
L
(
2
,
R
)
|
r
>
0
}
,
{\displaystyle \mathbf {A} =\left\{{\begin{pmatrix}r&0\\0&r^{-1}\end{pmatrix}}\in SL(2,\mathbb {R} )\ |\ r>0\right\},}
N
=
{
(
1
x
0
1
)
∈
S
L
(
2
,
R
)
|
x
∈
R
}
.
{\displaystyle \mathbf {N} =\left\{{\begin{pmatrix}1&x\\0&1\end{pmatrix}}\in SL(2,\mathbb {R} )\ |\ x\in \mathbf {R} \right\}.}
For the symplectic group G =Sp(2n , R ) , a possible Iwasawa decomposition is in terms of
K
=
S
p
(
2
n
,
R
)
∩
S
O
(
2
n
)
=
{
(
A
B
−
B
A
)
∈
S
p
(
2
n
,
R
)
|
A
+
i
B
∈
U
(
n
)
}
≅
U
(
n
)
,
{\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 ),}
A
=
{
(
D
0
0
D
−
1
)
∈
S
p
(
2
n
,
R
)
|
D
positive, diagonal
}
,
{\displaystyle \mathbf {A} =\left\{{\begin{pmatrix}D&0\\0&D^{-1}\end{pmatrix}}\in Sp(2n,\mathbb {R} )\ |\ D{\text{ positive, diagonal}}\right\},}
N
=
{
(
N
M
0
N
−
T
)
∈
S
p
(
2
n
,
R
)
|
N
upper triangular with diagonal elements = 1
,
N
M
T
=
M
N
T
}
.
{\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\}.}
Non-Archimedean Iwasawa decomposition
[ edit ]
There is an analog to the above Iwasawa decomposition for a non-Archimedean field
F
{\displaystyle F}
: In this case, the group
G
L
n
(
F
)
{\displaystyle GL_{n}(F )}
can be written as a product of the subgroup of upper-triangular matrices and the (maximal compact) subgroup
G
L
n
(
O
F
)
{\displaystyle GL_{n}(O_{F})}
, where
O
F
{\displaystyle O_{F}}
is the ring of integers of
F
{\displaystyle F}
.[2]
See also
[ edit ]
References
[ edit ]
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Iwasawa_decomposition&oldid=1116494073 "
C a t e g o r y :
● L i e g r o u p s
● T h i s p a g e w a s l a s t e d i t e d o n 1 6 O c t o b e r 2 0 2 2 , a t 2 1 : 3 1 ( U T C ) .
● T e x t i s a v a i l a b l e u n d e r t h e C r e a t i v e C o m m o n s A t t r i b u t i o n - S h a r e A l i k e L i c e n s e 4 . 0 ;
a d d i t i o n a l t e r m s m a y a p p l y . B y u s i n g t h i s s i t e , y o u a g r e e t o t h e T e r m s o f U s e a n d P r i v a c y P o l i c y . W i k i p e d i a ® i s a r e g i s t e r e d t r a d e m a r k o f t h e W i k i m e d i a F o u n d a t i o n , I n c . , a n o n - p r o f i t o r g a n i z a t i o n .
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