5 Invariant Koszul Form of Homogeneous Bounded Domains …
107
(θ ) = − log
∗
e
−θ,y dy = − log ψ (θ ) with θ ∈ sharp convex cone
ψ (θ ) =
e
−θ,y dy with Koszul-Vinberg Characteristic function
(5.32)
Jean-Louis Koszul has introduced the following forms
1st Koszul form α : α = d (θ ) = −d log ψ (θ )
(5.33)
2nd Koszul form γ : γ = Dα = Dd log ψ (θ )
(5.34)
with the following property of positive definitiveness:
(Dd log ψ Ω (x))(u) =
1
ψ Ω (u) 2
⎡
⎣
Ω ∗
F(ξ )
2 dξ.
Ω ∗
G(ξ )
2 dξ
−
⎛
⎝
Ω ∗
F(ξ ).G(ξ )dξ
⎞
⎠
2 ⎤
⎦
(Dd log ψ Ω (x))(u) > 0
with F(ξ ) = e
−
1
2
x,ξ and G(ξ ) = e
−
1
2
x,ξ
u, ξ
(5.35)
Koszul has defined the following Diffeomorphism:
η = α = −d log ψ (θ ) =
∗
ξ p θ (ξ )dξ with p θ (ξ ) =
e
−ξ,θ
e −ξ,θ dξ
(5.36)
with preservation of Legendre transform:
S (η) = =θ, η − (θ ) with η = d (θ ) and θ = d S (η)
(5.37)
This relations have been extended by Jean-Marie Souriau in geometric statistical mechanics, where he developed a “Lie groups thermodynamics” of dynamical
systems where the (maximum entropy) Gibbs density is covariant with respect to
the action of the Lie group [112–116]. In the Souriau model, previous structures of
information geometry are preserved:
2
( ),
2
*
( )
with ( )
log
and :
U
M
I
e
d
U M
ξ β
ω
β
β
λ
β
−
∂ Φ
= −
Φ
= −
∂
→
∫
g
(5.38)
We preserve the Legendre transform: