Page - 39 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 370 Proof. ForX,Y andZ inG,wehave sinceΘ is a 1-cocycle, ∑ circ(X,Y,Z) meaninga sumover circular permutationsofX,YandZ,usingthe Jacobi identity inG,wehave ∑ circ(X,Y,Z) 〈 Θb(X), [Y,Z] 〉 = ∑ circ(X,Y,Z) 〈−ad∗XEJ(b), [Y,Z]〉 = ∑ circ(X,Y,Z) 〈−EJ(b),[X, [Y,Z]]〉 =0. The linearmapΘb is thereforea1cocycle, evenasymplectic1-cocyclesince forallXandY∈G,〈 Θb(X),Y 〉 =−〈Θb(Y),X〉. Usingthefirstequalitystated inProposition17,wehaveforanyX∈G〈 Θb(b),X 〉 = 〈 Θ(b)−ad∗b EJ(b),X 〉 =−〈Θ(X),b〉+〈EJ(b), [X,b]〉=0. Ifwereplace Jby J1= J+μ, themapX →Θ(X) is replacedbyX →Θ1(X)=Θ(X)+ad∗Xμ andEJ(b)byEJ1(b)=EJ(b)+μ, thereforeΘb remainsunchanged. The following lemmawill allowus todefine, foreach b∈Ω, a remarkable symmetricbilinear formonthevectorsubspace [b,G]={[b,X] ;X∈G}of theLiealgebraG. Lemma1. LetΞbea1-cocycle of afinite-dimensionalLie algebraG for the coadjoint representation. For each b∈ kerΞ, let Fb = [G,b] be the set of elementsX∈Gwhich canbewrittenX= [X1,b] for someX1∈G. ThenFb is avector subspaceofG, and thevalueof the righthandsideof the equality Γb(X,Y)= 〈 Ξ(X1),Y 〉 , withX1∈G , X=[X1,b]∈Fb , Y∈Fb , dependsonlyonXandY,notonthechoiceofX1∈G such thatX=[X1,b]. That equalitydefinesabilinear formΓb onFbwhich is symmetric, i.e., satisfies Γb(X,Y)=Γb(Y,X) for allXandY∈Fb . Proof. LetX1 andX′1 ∈ G be such that [X1,b] = [X′1,b] =X. LetY1 ∈ G be such that [Y1,b] =Y. Wehave 〈 Ξ(X1−X′1),Y 〉 = 〈 Ξ(X1−X′1), [Y1,b] 〉 =−〈Ξ(Y1), [b,X1−X′1]〉−〈Ξ(b), [X1−X′1,Y1]〉 =0 sinceΞ(b)=0and [b,X1−X′1]=0.Wehaveshownthat 〈 Ξ(X1),Y 〉 = 〈 Ξ(X′1),Y 〉 . ThereforeΓb isa bilinear formonFb. Similarly〈 Ξ(X1),Y 〉 = 〈 Ξ(X1), [Y1,b] 〉 =−〈Ξ(Y1), [b,X1]〉−〈Ξ(b), [X1,Y1]〉= 〈Ξ(Y1),X〉 , whichproves thatΓb is symmetric. Theorem7. Theassumptionsandnotationsarethesameasthose inProposition14. Foreachb∈Ω, thereexists on the vector subspace Fb = [G,b] of elementsX∈Gwhich can bewrittenX= [X1,b] for someX1 ∈G, a symmetricnegativebilinear formΓb givenby Γb(X,Y)= 〈 Θb(X1),Y 〉 , withX1∈G , X=[X1,b]∈Fb , Y∈Fb , 39