Page - 124 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 254 Taking intoaccount (4),onefinds that thecoadjoint representation: Ad∗(a) :g∗→g∗ :μ′ →μ=Ad∗(a)μ′ isgivenby: K=K′P−1, L=(PL′+CK′)P−1 . It isnoteworthytoobserve that the transformation law(3)ofmomenta isnothingother thanthe coadjoint representation! However, thismathematical construction isnot relevant forall consideredphysicalapplications andweneedtoextenditbyconsideringamapθ fromG intog∗ andageneralizedtransformation law: μ= a ·μ′=Ad∗(a)μ′+θ(a) , (6) where θ eventually depends on an invariant of the orbit. It is an affine representation ofG in g∗ (becausewewishthemomentumtobeanaffinetensor)provided: ∀a,b∈G, θ(ab)= θ(a)+Ad∗(a)θ(b) (7) Remark1. This action inducesa structureof affine spaceon the set ofmomentumtensors. Letπ :F→Mbe aG-principal bundleof affine frameswith the freeaction (a, f) → f ′= a · f oneachfiber. Thenwecanbuild theassociatedG-principal bundle: πˆ :g∗×F→ (g∗×F)/G : (μ, f) →μ= orb(μ, f) for the free action: (a,(μ, f)) → (μ′, f ′)= a ·(μ, f)=(a ·μ,a · f) where theactionong∗ is (6). Clearly, theorbitμ= orb(μ, f) canbe identified to themomentumG-tensorμof componentsμ in theG-frame f. 4. SymplecticActionandMomentumMap Let (N ,ω)bea symplecticmanifold [3,4,6,10]. ALiegroupG smoothly left actingonN and preserving thesymplectic formω is said tobesymplectic.The interiorproductofavector −→ V anda p-formω isdenoted ι( −→ V )ω. Amapψ :N→g∗ suchthat: ∀η∈N , ∀Z∈g, ι(Z ·η)ω=−d(ψ(η)Z) , is calledamomentummapofG. It is thequantity involved inNoether’s theoremthat claimsψ is constantoneach leafofN . In [3] (Theorem11.17,p. 109,or itsEnglish translation[4]), Souriauproved thereexistsasmoothmapθ fromG intog∗: θ(a)=ψ(a ·η)−Ad∗(a)ψ(η) , (8) which isasymplecticcocycle, that isamapθ :G→gverifyingthe identity (7)andsuchthat (Dθ)(e) is a 2-form. An important result, called theKirillov–Kostant–Souriau theorem, reveals the orbit symplectic structure [3] (Theorem11.34, Pages 116–118). LetGbeaLiegroupandanorbit of the coadjointrepresentationorb(μ)⊂g∗. Thentheorbitorb(μ) isasymplecticmanifold,G isasymplectic groupandanyμ∈g∗ is itsownmomentum. Remark2. Replacingη by a−1 ·η in (8), this formula reads: ψ(η)=Ad∗(a)ψ′(η)+θ(a) , 124