Page - 38 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 370 Then for eachb∈Ωandeachg∈G P(Adgb)= exp (〈 θ(g−1),b 〉) P(b)= exp ( −〈Ad∗gθ(g),b〉)P(b) , EJ(Adgb)=Ad∗g−1 EJ(b)+θ(g) , S(Adgb)=S(b) . Proof. Wehave P(Adgb)= ∫ M exp (−〈J,Adgb〉)dλω= ∫ M exp (−〈Ad∗g J,b〉)dλω = ∫ M exp ( −〈J◦Φg−1−θ(g−1,b〉)dλω = exp (〈 θ(g−1),b 〉) P(b)= exp ( −〈Ad∗gθ(g),b〉)P(b) , sinceθ(g−1)=−Ad∗gθ(g). ByusingPropositions14and12, theotherresultseasily follow. Remark16. Theequality EJ(Adgb)=Ad∗g−1 EJ(b)+θ(g) means that themapEJ :Ω→G∗ is equivariantwith respect to theadjoint actionofGontheopensubsetΩof itsLie algebraG and its affineactiononthe left onG∗ (g,ξ) →Ad∗g−1 ξ+θ(g) , g∈G , ξ∈G∗ . Proposition17. Theassumptionsandnotationsare the sameas those inProposition14. For eachb∈Ωand eachX∈G,wehave 〈 EJ(b), [X,b] 〉 = 〈 Θ(X),b 〉 , DEJ(b) ( [X,b] ) =−ad∗XEJ(b)+Θ(X) , whereΘ=Teθ :G→G∗ is the1-cocycle of theLie algebraG associated to the1-cocycleθ of theLiegroupG. Proof. Letussetg= exp(τX) in thefirstequality inProposition16,derive thatequalitywithrespect toτ, andevaluate theresultatτ=0.Weobtain DP(b) ( [X,b] ) =−P(b)〈Θ(X),b〉 . Since,bythefirstequalityofProposition14,DP(b)=−P(b)EJ(b), thefirststatedequalityfollows. Letusnowsetg= exp(τX) in thesecondequality inProposition16,derive thatequalitywith respect toτ, andevaluate theresultatτ=0.Weobtain thesecondequalitystated. Corollary4. With theassumptionsandnotationsofProposition17, letusdefine, for eachb∈Ω, a linearmap Θb :G→G∗ bysetting Θb(X)=Θ(X)−ad∗XEJ(b) . ThemapΘb is a symplectic1-cocycle of theLie algebraG for the coadjoint representation,whichsatisfies Θb(b)=0. Moreover ifwe replace themomentummap J by J1 = J+μ, withμ∈ G∗ constant, the1-cocycleΘb remainsunchanged. 38