Page - 72 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 386 Θ˜(X,Y) : g×g→ (81) isaskew-symmetricbilinear form,andiscalledthesymplecticCocycleofLiealgebragassociatedto themomentmap J. LetΘ : g→ g∗ bethemapsuchthat forall: X,Y∈ g : 〈Θ(X),Y〉= Θ˜(X,Y) (82) ThemapΘ is therefore theone-cocycleof theLiealgebra gwithvalues in g∗ for the coadjoint representation X → ad∗X ofgassociatedto theaffineactionofgonitsdual: aΘ(X)(ξ)= ad∗−X(ξ)+Θ(X) , X∈ g , ξ∈ g∗ (83) LetGbeaLiegroupwhoseLiealgebra isg. Theskew-symmetricbilinear form Θ˜ong=TeG can beextendedintoacloseddifferential two-formonG, since the identityon Θ˜meansthat itsexterior differentialdΘ˜vanishes. Inotherwords, Θ˜ isa2-cocycle for therestrictionof thedeRhamcohomology ofG to left (or right) invariantdifferential forms. 6.3. EquivarianceofSouriauMomentMap Thereexistsauniqueaffineaction a suchthat the linearpart isacoadjoint representation: a :G×g∗→ g∗ a(g,ξ)=Ad∗g−1ξ+θ(g) (84) with 〈 Ad∗g−1ξ,X 〉 = 〈 ξ,Adg−1X 〉 andthat induceequivarianceofmoment J. 6.4.ActionofLieGrouponaSymplecticManifold LetΦ :G×M→M beanactionofLiegroupGondifferentiablemanifoldM, the fundamental fieldassociatedtoanelementXofLiealgebragofgroupG is thevectorsfieldXMonM: XM(x)= d dt Φexp(−tX) (x) ∣∣∣∣ t=0 WithΦg1 ( Φg2(x) ) =Φg1g2(x)andΦe(x)= x (85) Φ isHamiltonianonasymplecticmanifoldM, ifΦ is symplecticandif forallX∈ g, the fundamental fieldXM isgloballyHamiltonian. ThereisauniqueactionaoftheLiegroupGonthedualg∗of itsLiealgebraforwhichthemoment map J isequivariant, thatmeanssatisfies foreachx∈M J ( Φg(x) ) = a(g, J(x))=Ad∗g−1 (J(x))+θ(g) (86) θ :G→ g∗ is calledcocycleassociated to thedifferentialTeθof1-cocyle θ associated to Jatneutral element e: 〈Teθ(X),Y〉= Θ˜(X,Y)= J[X,Y]−{JX, JY} (87) If insteadof Jwetakethemomentmap J′(x)= J(x)+μ , x∈M,whereμ∈ g∗ is constant, the symplecticcocycleθ is replacedby: θ′(g)= θ(g)+μ−Ad∗gμ (88) whereθ′−θ=μ−Ad∗gμ isone-coboundaryofGwithvalues ing∗. 72