Page - 18 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 370 with the following convention: ψ is a Lie algebras homomorphismwhenwe take for Lie algebraG of theLie groupGtheLiealgebraor right invariantvectorfieldsonGifΨ is anactiononthe left, and theLiealgebraof left invariantvectorfieldsonGifΨ is anactiononthe right. Proof. IfΨ isanactionofGonMonthe left (respectively,on theright), thevectorfieldonGwhich is right invariant (respectively, left invariant)andwhosevalueat e isX, andtheassociatedfundamental vectorfieldXMonM, arecompatiblebythemapg →Ψg(x). Therefore themapψ :G→A1(M) isa Liealgebrashomomorphism, ifwetake fordefinitionof thebracketonG thebracketof right invariant (respectively, left invariant)vectorfieldsonG. Definition12. WhenMisapresymplectic (or a symplectic, or aPoisson)manifold, anactionΨof aLiegroup G (respectively, anactionψ of aLie algebraG) on themanifoldMis called apresymplectic (or a symplectic, or aPoisson)action if for eachg∈G,Ψg is apresymplectic, or a symplectic, or aPoissondiffeomorphismofM (respectively, if for eachX∈G,ψ(X) is apresymplectic, or a symplectic, or aPoissonvectorfieldonM. Definition13. AnactionψofaLiealgebaG onapresymplecticorsymplecticmanifold(M,ω), oronaPoisson manifold (M,Λ), is said tobeHamiltonian if for eachX∈G, thevectorfieldψ(X)onMisHamiltonian. AnactionΨ (eitheron the left or on the right) of aLiegroupGonapresymplectic or symplecticmanifold (M,ω), oronaPoissonmanifold(M,Λ), is saidtobeHamiltonian if thatactionispresymplectic, orsymplectic, orPoisson (according to the structureofM), and if inaddition theassociatedactionof theLiealgebraG ofG isHamiltonian. Remark9. AHamiltonianactionof aLiegroup, orof aLie algebra, onapresymplectic, symplectic orPoisson manifold, is automatically a presymplectic, symplectic or Poisson action. This result immediately follows fromProposition3. 5.3.MomentumMapsofHamiltonianActions Proposition 5. Let ψ be a Hamiltonian action of a finite-dimensional Lie algebra G on a presymplectic, symplectic orPoissonmanifold (M,ω)or (M,Λ). There exists a smoothmap J :M→G∗, taking itsvalues in thedual spaceG∗ of theLie algebraG, such that for eachX∈G theHamiltonianvectorfieldψ(X)onMadmits asHamiltonian the function JX :M→R, definedby JX(x)= 〈 J(x),X 〉 , x∈M . Themap J is calledamomentummap for theLie algebraactionψ.Whenψ is theactionof theLie algebra G of aLiegroupGassociated toaHamiltonianactionΨof aLiegroupG, J is calledamomentummap for the HamiltonianLiegroupactionΨ. Theproofof that result,which iseasy, canbefoundforexample in [8–10]. Remark10. Themomentummap J isnotunique: • when (M,ω) is a connectedsymplecticmanifold, J isdetermineduptoadditionof anarbitraryconstant element inG∗; • when (M,Λ) is a connectedPoissonmanifold, themomentummap J is determinedup to additionof an arbitraryG∗-valuedsmoothmapwhich, coupledwithanyX∈G, yieldsaCasimirof thePoissonalgebra of (M,Λ), i.e., a smooth function onMwhosePoisson bracketwith any other smooth function on that manifold is the function identically equal to0. 5.4.Noether’sTheoreminHamiltonianFormalism Theorem5 (Noether’sTheoreminHamiltonianFormalism). LetXf andXg be twoHamiltonianvector fields on a presymplectic or symplecticmanifold (M,ω), or on aPoissonmanifold (M,Λ), which admit as 18