Page - 98 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 386 WecanrewriteAdMβwiththe following identification: AdMβ= ⎡⎢⎣ 12Ω−1 −Ω−1n 0 0 ⎤⎥⎦ withΩ=R′1/2RR′−1/2 andn= ( 1 2 m′+R′1/2m ) (249) Wehave then todevelop ξˆ(AdM(β)), that is to say ξˆ(β) after action of the groupon theLie algebra forβ, givenbyAdM(β). Byanalogyofstructurebetween ξˆ(β)andβ,wecanwrite: β= ⎡⎣ 12R−1 −R−1m 0 0 ⎤⎦ ξˆ(β)= [ R+mmT m 0 0 ] ⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭ ⇒ ⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩ AdMβ= ⎡⎣ 12Ω−1 −Ω−1n 0 0 ⎤⎦ ξˆ(AdM(β))= [ Ω+nnT n 0 0 ] (250) We have then to identify the cocycle θ(M) from ξˆ(AdM(β)) = Ad∗M(ξˆ) + θ(M) ⇒ θ(M)= ξˆ(AdM(β))−Ad∗Mξˆ where: Ad∗Mξˆ= [ R+mmT−mm′T R′1/2m 0 0 ] (251) ξˆ(AdM(β))= ⎡⎣ R′1/2RR′−1/2+(12m′+R′1/2m)(12m′+R′1/2m)T (12m′+R′1/2m) 0 0 ⎤⎦ (252) Thecocycle is thengivenby: θ(M)= ⎡⎣ R′1/2RR′−1/2+(12m′+R′1/2m)(12m′+R′1/2m)T (12m′+R′1/2m) 0 0 ⎤⎦−[ R+mmT−mm′T R′1/2m 0 0 ] θ(M)= ⎡⎢⎣ ( R′1/2RR′−1/2−R ) + ( R′1/2mmTR′1/2T−mmT ) + ( 1 2m ′mTR′1/2T+ 1 2 R′1/2mm′T−mm′T ) 1 2m ′ 0 0 ⎤⎥⎦ (253) Fromθ(M)= ξˆ(AdM(β))−Ad∗Mξˆ,wecancomputecocycle inLiealgebra Θ=Teθ (254) usedtodefinethe tensor: Θ˜(X,Y) : g×g→ X,Y → 〈Θ(X),Y〉 (255) In this secondpart,wewill compute theSouriau-Fishermetricgivenby: gβ([β,Z1] , [β,Z2])= Θ˜β(Z1, [β,Z2]) (256) with Θ˜β(Z1,Z2)= Θ˜(Z1,Z2)+ 〈 ξˆ,adZ1Z2 〉 = 〈Θ(Z1),Z2〉+ 〈 ξˆ, [Z1,Z2] 〉 (257) gβ([β,Z1] , [β,Z2])= Θ˜β(Z1, [β,Z2])= Θ˜(Z1, [β,Z2])+ 〈 ξˆ, [Z1, [β,Z2]] 〉 = 〈Θ(Z1) , [β,Z2]〉+ 〈 ξˆ, [Z1, [β,Z2]] 〉 (258) 98