Page - 70 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 386 δ t1 t0 Φ(β(t)) ·dt=0 (66) 6. SouriauAffineRepresentationofLieGroupandLieAlgebraandComparisonwiththeKoszul AffineRepresentation Thisaffine representationofLiegroup/algebrausedbySouriauhasbeen intensively studied byMarle [7,100,109,110]. Souriaucalledthemechanicsdeducedfromthismodel, “affinemechanics”. Wewill explainaffinerepresentationsandassociatednotionsascocycles,Souriaumomentmapand cocycles, equivarianceofSouriaumomentmap,actionofLiegrouponasymplecticmanifoldanddual spacesoffinite-dimensionalLiealgebras.Wehaveobservedthat these toolshavebeendevelopedin parallelbyJean-LouisKoszul.Wewill establishclose linksandsynthetize thecomparisons ina table ofbothapproaches. 6.1.AffineRepresentationsandCocycles SouriaumodelofLiegroupthermodynamics is linkedwithaffinerepresentationofLiegroup andLiealgebra.Wewillgive in the followingmainelementsof thisaffinerepresentation. LetGbeaLiegroupandEafinite-dimensionalvectorspace.Amap A :G→Af f(E) canalways bewrittenas: A(g)(x)=R(g)(x)+θ(g)withg∈G,x∈E (67) where the maps R :G→GL(E) and θ :G→E are determined by A. The map A is an affine representationofG inE. Themap θ :G→E isaone-cocycleofGwithvalues inE, for the linearrepresentationR; itmeans thatθ isasmoothmapwhichsatisfies, forallg,h∈G: θ(gh)=R(g)(θ(h))+θ(g) (68) The linearrepresentationR is calledthe linearpartof theaffinerepresentationA, andθ is calledthe one-cocycleofGassociatedto theaffinerepresentationA.Aone-coboundaryofGwithvalues inE, for the linearrepresentationR, isamap θ :G→E whichcanbeexpressedas: θ(g)=R(g)(c)−c , g∈G (69) wherec isafixedelement inEandthenthereexistanelement c∈Esuchthat, forallg∈Gandx∈E: A(g)(x)=R(g)(x+c)−c (70) Let gbe aLie algebra andE afinite-dimensional vector space. A linearmap a : g→ af f(E) alwayscanbewrittenas: a(X)(x)= r(X)(x)+Θ(X)withX∈ g,x∈E (71) where the linearmaps r : g→ gl(E) and Θ : g→Eare determined by a. Themap a is an affine representationofG inE.The linearmapΘ : g→E isaone-cocycleofGwithvalues inE, for the linear representation r; itmeans thatΘ satisfies, forallX,Y∈ g: Θ([X,Y])= r(X)(Θ(Y))−r(Y)(Θ(X)) (72) Θ is calledtheone-cocycleofgassociatedto theaffinerepresentationa.Aone-coboundaryofgwith values inE, for the linear representation r, is a linearmap Θ : g→E which can be expressed as: Θ(X)= r(X)(c) , X∈ gwhere c isafixedelement inE., andthenthereexistanelement c∈E such that, forallX∈ gandx∈E: a(X)(x)= r(X)(x+c) 70