Page - 74 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 386 ThisPoissonstructureiscalledthemodifiedcanonicalPoissonstructurebymeansofthesymplectic cocycle Θ˜. Thesymplectic leavesofg∗ equippedwiththisPoissonstructureare theorbitsofanaffine actionwhose linearpart is thecoadjointaction,withanadditional termdeterminedby Θ˜. 6.6. KoszulAffineRepresentationofLieGroupandLieAlgebra Previously,wehavedevelopedSouriau’sworksontheaffinerepresentationofaLiegroupusedto elaborate theLiegroupthermodynamics.Wewill studyhereanotherapproachofaffinerepresentation ofLiegroupandLiealgebra introducedbyJean-LouisKoszul.Weconsolidate the linkof Jean-Louis KoszulworkwithSouriaumodel. ThismodelusesanaffinerepresentationofaLiegroupandofaLie algebra inafinite-dimensionalvectorspace, seenasspecialexamplesofactions. Since theworkofHenriPoincareandElieCartan, the theoryofdifferential formshasbecome an essential instrument of modern differential geometry [112–115] used by Jean-Marie Souriau for identifying the space of motions as a symplectic manifold. However, as said by Paulette Libermann [116], exceptHenri Poincaréwhowrote shortly before his death a report on thework ofElieCartanduringhis application for the SorbonneUniversity, theFrenchmathematiciansdid notsee the importanceofCartan’sbreakthroughs. SouriaufollowedlecturesofElieCartan in1945. ThesecondstudentofElieCartanwasJean-LouisKoszul.Koszul introducedtheconceptsofaffine spaces,affine transformationsandaffinerepresentations [117–124].Moreespecially,weare interested byKoszul’s definition for affine representations of Lie groups and Lie algebras. Koszul studied symmetrichomogeneousspacesanddefinedrelationbetweeninvariantflataffineconnectionstoaffine representationsofLiealgebras,andcharacterized invariantHessianmetricsbyaffinerepresentations ofLiealgebras [117–124].Koszulprovidedcorrespondencebetweensymmetrichomogeneousspaces with invariantHessianstructuresbyusingaffinerepresentationsofLiealgebras,andprovedthata simplyconnectedsymmetrichomogeneousspacewith invariantHessianstructure isadirectproduct ofaEuclideanspaceandahomogeneousself-dualregularconvexcone[117–124]. LetGbeaconnected LiegroupandletG/KbeahomogeneousspaceonwhichGactseffectively,Koszulgaveabijective correspondencebetweenthesetofG-invariantflatconnectionsonG/Kandthesetofacertainclass of affine representationsof theLie algebraofG [117–124]. Themain theoremofKoszul is: letG/K beahomogeneousspaceofaconnectedLiegroupGandlet gandkbe theLiealgebrasofGandK, assumingthatG/K isendowedwithaG-invariantflatconnection, thengadmitsanaffinerepresentation (f,q)on thevectorspaceE.Conversely, suppose thatG is simplyconnectedandthatg is endowedwith anaffinerepresentation, thenG/KadmitsaG-invariantflatconnection. Koszul has proved the following [117–124]. LetΩ be a convex domain inRn containing no complete straight lines, andanassociatedconvexconeV(Ω)= {(λx,x)∈Rn×R/x∈Ω,λ∈R+}. Thenthereexistsanaffineembedding: : x∈Ω → [ x 1 ] ∈V(Ω) (94) Ifweconsiderη thegroupofhomomorphismofA(n,R) intoGL(n+1,R)givenby: s∈A(n,R) → [ f(s) q(s) 0 1 ] ∈GL(n+1,R) (95) andassociatedaffinerepresentationofLiealgebra:[ f q 0 0 ] (96) withA(n,R) thegroupofallaffinetransformationsofRn.Wehaveη(G(Ω))⊂G(V(Ω))andthepair (η, )of thehomomorphism η :G(Ω)→G(V(Ω)) andthemap :Ω→V(Ω) isequivariant. 74