5 Invariant Koszul Form of Homogeneous Bounded Domains …
109
[ ] {
}
,
( , )
,
X
Y
X Y
X Y
J
J J
Θ
=
−

with ( ) :
such that ( )
( ), ,
X
J x M
J x
J x X X
→
=
∈
*
g
g
(5.44)
This tensor ˜
is also defined in tangent space of the cocycle ( )
g
θ
∈
*
g (this
cocycle appears due to the non-equivariance of the coadjoint operator Ad
∗
g , action
of the group on the dual space of the lie algebra; the action of the group on the dual
space of the Lie algebra is modified with a cocycle so that the momentu map becomes
equivariant relative to this new affine action):
Q
Ad g (β)
= Ad
∗
g (Q) + θ(g)
(5.45)
( )
g
θ
∈
*
g is called nonequivariance one-cocycle, and it is a measure of the lack of
equivariance of the moment map.
( )
(
)
, :
with ( )
( )
X,Y
( ),
e
X Y
X T X e
X Y
θ
Θ
× →ℜ
Θ
=
Θ

:
g g
(5.46)
Souriau has then defined a Gibbs density that is covariant under the action of the
group:
p Gibbs (ξ ) = e
(ξ ),β
=
e
−U (ξ ),β
M e −U (ξ ),β dλ ω
with (β) = − log
M
e
−−U (ξ ),β dλ ω
(5.47)
Q =
∂∂(β)
∂β
=
M
U (ξ )e
−U (ξ ),β dλ ω
M
e −U (ξ ),β dλ ω
=
M
U (ξ ) p(ξ )dλ ω
(5.48)
We can express the Gibbs density with respect to Q by inverting the relation
Q =
∂∂(β)
∂β
=
Then p Gibbs,Q (ξ ) = e
(β)− (ξ ),,
−1 (Q) with β =
−1
(Q).
This domain is very active and close approaches are explored by Koichi Tojo
to build harmonic exponential families on homogeneous spaces [117, 118, 115]
(Figs. 5.6 and 5.7).
We will introduce Souriau moment map for SU(1,1)/U(1) group that acts
transitively on Poincaré Unit Disk, based on moment map. Considering the Lie
group
SU (1, 1) =
a b
b
∗ a
∗
/a, b ∈ C, |a|
2
− |b|
2
= 1
(5.35)