Page - 91 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 386 ButasΠL=nL,wecandeduce that:〈 nL,M−1nRM 〉 = 〈ΠR,nR〉 withM= [ R1/2 m 0 1 ] , nL= [ R−1/2 . R 1/2 R−1/2 .m 0 0 ] andηR= [ R−1/2 . R 1/2 . m−R−1/2 .R1/2 .m 0 0 ] ⇒ΠR= [ R−1/2 . R 1/2 +R−1 .mmT R−1 .m 0 0 ] (208) Then, theoperator that transformtherightalgebra to itsdualalgebra isgivenby: βM : g→ g∗ nR= [ ηR1 ηR2 0 0 ] →ΠR= [ ηR1 ( 1+mTR−1m ) +ηR2mTR−1 ηR1R−1m+R−1ηR2 0 0 ] (209) There isanoperator tochangetheviewofalgebra. Therefore, there isanoperator thatdid the sameto thedualalgebra. This is called theco-adjointoperatorandit is theconjugateactionof theLie grouponitsdualalgebra:{ Ad∗ :G×g∗→ g M,η →Ad∗Mη with 〈Ad∗Mη,n〉= 〈η,AdMn〉wheren∈ g (210) Wecanthendevelopthisexpressionforouruse in thecaseofanaffinesup-group.Wefind: ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ M= [ A b 0 1 ] ∈G η= [ η1 η2 0 0 ] ∈ g∗ n= [ n1 n2 0 0 ] ∈ g ⇒ ⎧⎪⎪⎨⎪⎪⎩ 〈 Ad∗Mη,n 〉 = 〈η,AdMn〉= 〈 η,MnM−1 〉 〈 Ad∗Mη,n 〉 = 〈[ η1−η2bT Aη2 0 0 ] ,n 〉 ⇒Ad∗Mη= [ η1−η2bT Aη2 0 0 ] (211) andwecanalsoobserve that: Ad∗M−1η= [ η1+Aη2bT Aη2 0 0 ] (212) Similarly thereexists the followingrelationbetweenthe leftandtherightalgebras: Ad∗MΠR=ΠL andAd∗M−1ΠL=ΠR (213) Aswehavedefineda commutator on theLie algebra, it is possible todefineoneon its dual algebra. Thiscommutatoronthedualalgebracanalsobedefinedusingtheoperatorexpressingthe combinedactionof thealgebraof itsdualalgebra. Thisoperator iscalledtheco-adjointoperator:{ ad∗ : g×g∗→ g∗ n,η → ad∗nη with 〈ad∗nη,κ〉= 〈η,adnκ〉whereκ∈ g (214) Wecandevelopthisco-adjointoperatoron itsdualalgebra forouruse-case: 91