Page - 15 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 370 Definition 6. APoissonmanifold is a smoothmanifold Pwhose algebra of smooth functionsC∞(P,R) is endowedwithabilinearcomposition law,called thePoissonbracket,whichassociates toanypair(f,g)of smooth functionsonPanother smooth functiondenotedby{f,g}, that compositionsatisfying the threeproperties 1. it is skew-symmetric, {g, f}=−{f,g}, 2. it satisfies the Jacobi identity{ f,{g,h}}+{g,{h, f}}+{h,{f,g}}=0, 3. it satisfies theLeibniz identity {f,gh}={f,g}h+g{f,h}. Example4. 1. Onthevectorspaceofsmooth functionsdefinedonasymplecticmanifold(M,ω), thereexistsacomposition law, called thePoissonbracket,which satisfies theproperties stated inDefinition6. Letus recall briefly its definition. The symplectic formω allowsus toassociate, to anysmooth function f ∈C∞(M,R), a smooth vectorfieldXf ∈A1(M,R), called theHamiltonianvectorfieldassociated to f, definedby i(Xf)ω=−df . The Poisson bracket {f,g} of two smooth functions f and g ∈ C∞(M,R) is defined by the three equivalent equalities {f,g}= i(Xf)dg=−i(Xg)df=ω(Xf,Xg) . Anysymplecticmanifold is therefore aPoissonmanifold. ThePoissonbracket of smooth functionsdefinedonasymplecticmanifold (whenthat symplecticmanifold is a cotangentbundle)wasdiscoveredbySiméonDenisPoisson (1781–1840) [39]. 2. LetG beafinite-dimensional realLie algebra, and letG∗ be itsdualvector space. For eachsmooth function f ∈C∞(G∗,R)andeachζ∈G∗, thedifferentialdf(ζ) is a linear formonG∗, inotherwordsanelementof thedualvector spaceofG∗. IdentifyingwithG thedualvector spaceofG∗,wecan therefore considerdf(ζ) asanelement inG.With this identification,wecandefine thePoissonbracket of twosmooth functions f andg∈C∞(G∗,R)by {f,g}(ζ)= [df(ζ),dg(ζ)] , ζ∈G∗ , thebracket in the righthandsidebeing thebracket in theLiealgebraG. ThePoissonbracket of functions in C∞(G∗,R) so defined satifies the properties stated in Definition 6. The dual vector space of any finite-dimensional real Lie algebra is therefore endowedwith a Poisson structure, called its canonical Lie-Poisson structure or its Kirillov-Kostant-Souriau Poisson structure. Discovered by Sophus Lie, this structure was indeed rediscovered independently by Alexander Kirillov, Bertram Kostant and Jean-MarieSouriau. 3. A symplectic cocycle of a finite-dimensional, real Lie algebraG is a skew-symmetric bilinearmapΘ : G×G→G∗whichsatisfies, for allX,YandZ∈G, Θ ( [X,Y],Z ) +Θ ( [Y,Z],X ) +Θ ( [Z,X],Y ) =0. ThecanonicalLie-Poissonbracket of twosmooth functions f andg∈C∞(G∗,R) canbemodifiedbymeans of the symplectic cocycleΘ, by setting {f,g}Θ(ζ)= [ df(ζ),dg(ζ) ]−Θ(df(ζ),dg(ζ)) , ζ∈G∗ . 15