Seite - 89 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 386 theautomorphisms, theactionbyconjugationof theLiegrouponitself thatallowsthisoperator to carryamemberof thegroup. AD :G×G→G M,N →ADMN=M.N.M−1 (192) ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩ M1= [ R1/21 m1 0 1 ] , M2= [ R1/22 m2 0 1 ] ADM1M2= [ R1/22 −R1/22 m1+R1/21 m2+m1 0 1 ] (193) IfnowweconsideracurveN(t)curveonthemanifoldvia the identityat t=0. Its imagebythe previousoperatorwill be thencurveγ=M ·N(t) ·M−1 passing through identityelementat t=0. As . N(0) isanelementof theLiealgebraandits imagebypreviousconjugationoperator iscalledthe Adjointoperator: Ad :G×g→ g M,n →AdMn=M.n.M−1= ddt ∣∣∣ t=0 (ADMN(t))with { N(0)= I . N(0)=n∈ g (194) Wecanthencompute theAdjointoperator for thepreviousLiegroup: ⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩ n2L= ⎡⎣ R−1/22 .R1/22 R−1/22 .m2 0 0 ⎤⎦ , n2R= ⎡⎣ R−1/22 .R1/22 −R−1/22 .R1/22 m2+ .m2 0 0 ⎤⎦ AdM1n2L=n2R andAdM2n2R= ⎡⎣ R−1/22 .R1/22 −R−1/22 .R1/22 m2+ .R1/22 m2+R1/22 .m2 0 0 ⎤⎦ , AdM−11 n2R=n2L (195) Werecall that theLiealgebrahasbeendefinedas the tangentspaceat the identityofaLiegroup. Wewill then introduceaLiebracket [., .], theexpressionof theoperatorassociatedwith thecombined actionof theLiealgebraon itself, calledanadjointoperator. Theadjointoperator represents theaction byconjugationof theLiealgebraonitselfandisdefinedby: ad : g×g→ g n,m → admn=m ·n−n ·m= ddt ∣∣∣ t=0 (AdMn(t))= [m,n]with { . N(0)=n∈ g . M(0)=m∈ g (196) Wecanthencompute thisoperator forourusecase: n1L= ⎡⎣ R−1/21 .R1/21 R−1/21 .m1 0 0 ⎤⎦ , n2L= ⎡⎣ R−1/22 .R1/22 R−1/22 .m2 0 0 ⎤⎦ (197) adn1Ln2L=[n1L,n2L]= ⎡⎢⎣ 0 R−1/21 ( . R 1/2 1 . m2− . R 1/2 2 . m1 ) R−1/22 0 0 ⎤⎥⎦ (198) adn1Rn2R=[n1R,n2R]= ⎡⎣ 0 R−1/21 .R1/21 ( −R−1/22 . R 1/2 2 m2+ . m2 ) −R−1/22 . R 1/2 2 ( −R−1/21 . R 1/2 1 m1+ . m1 ) 0 0 ⎤⎦ (199) 89