Page - 71 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 386 Let A :G→Af f(E) beanaffinerepresentationofaLiegroupg inafinite-dimensionalvector space E, and g be the Lie algebra of G. Let R :G→GL(E) and θ :G→E be, respectively, the linearpartandtheassociatedcocycleof theaffinerepresentationA.Let a : g→ af f(E) betheaffine representation of the Lie algebra g associated to the affine representation A :G→Af f(E) of the LiegroupG. The linearpart of a is the linear representation r : g→ gl(E) associated to the linear representation R :G→GL(E), and the associated cocycleΘ : g→E is related to the one-cocycle θ :G→E by: Θ(X)=Teθ(X(e)) ,X∈ g (73) This isdeducedfrom: dA(exp(tX))(x) dt ∣∣∣∣ t=0 = d(R(exp(tX))(x)+θ(exp(tX)) dt ∣∣∣∣ t=0 ⇒ a(X)(x)= r(X)(x)+Teθ(X) (74) LetGbeaconnectedandsimplyconnectedLiegroup, R :G→GL(E) bea linear representation ofG inafinite-dimensionalvectorspaceE, and r : g→ gl(E) betheassociated linearrepresentation of itsLiealgebrag. Foranyone-cocycleΘ : g→E of theLiealgebrag for the linearrepresentation r, thereexistsauniqueone-cocycle θ :G→E oftheLiegroupG for thelinearrepresentationRsuchthat: Θ(X)=Teθ(X(e)) (75) inotherwords,whichhasΘasassociatedLiealgebraone-cocycle. TheLiegroupone-cocycleθ isaLie groupone-coboundary ifandonly if theLiealgebraone-cocycleΘ isaLiealgebraone-coboundary. dθ(gexp(tX)) dt ∣∣∣∣ t=0 = d(θ(g)+R(g)(θ(exp(tX))) dt ∣∣∣∣ t=0 ⇒Tgθ ( TLg(X) ) =R(g)(Θ(x)) (76) whichproves that if it exists, theLiegroupone-cocycleθ suchthatTeθ=Θ isunique. 6.2. SouriauMomentMapandCocycles Souriaufirst introducedthemomentmapinhisbook.Wewillgivethelinkwithpreviouscocycles ofaffinerepresentation. Thereexist JX linearapplicationfromg todifferential functiononM: g→C∞(M,R) X→ JX (77) Wecanthenassociateadifferentiableapplication J, calledmoment(um)mapfor theHamiltonian LiegroupactionΦ: J :M→ g∗ x → J(x) suchthat JX(x)= 〈J(x),X〉 , X∈ g (78) Let Jmomentmap, foreach (X,Y)∈ g×g,weassociateasmooth function Θ˜(X,Y) :M→ definedby: Θ˜(X,Y)= J[X,Y]−{JX, JY}with {., .} : PoissonBracket (79) It isaCasimirof thePoissonalgebraC∞(M, ), that satisfies: Θ˜([X,Y] ,Z)+Θ˜([Y,Z] ,X)+Θ˜([Z,X] ,Y)=0 (80) WhenthePoissonmanifold isaconnectedsymplecticmanifold, the function Θ˜(X,Y) is constant onMandthemap: 71