Page - 77 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 386 Ifweconsiderη thegroupofhomomorphismofA(n,R) intoGL(n+1,R)givenby: s∈A(n,R) → [ f(s) q(s) 0 1 ] ∈GL(n+1,R) (110) withA(n,R) thegroupofall affine transformationsofRn. Wehaveη(G(Ω))⊂G(V(Ω))and the pair (η, )of thehomomorphism η :G(Ω)→G(V(Ω)) andthemap :Ω→V(Ω) isequivariant: ◦s=η(s)◦ andd ◦s=η(s)◦d (111) 6.7. ComparisonofKoszul andSouriauAffineRepresentationofLieGroupandLieAlgebra Wewill compare, in the followingTable1,affinerepresentationofLiegroupandLiealgebra from SouriauandKoszulapproaches: Table1.TablecomparingSouriauandKoszulaffinerepresentationofLiegroupandLiealgebra. SouriauModelofAffineRepresentationofLie GroupsandAlgebra KoszulModelofAffineRepresentationofLie GroupsandAlgebra A(g)(x)=R(g)(x)+θ(g)withg∈G,x∈E R :G→GL(E) and θ :G→E Af f(s) : a → sa= f(s)a+q(s) ∀s∈G,∀a∈E f :G→GL(E) s → f(s)a= sa−so ∀a∈E q :G→E s →q(s)= so ∀s∈G θ(gh)=R(g)(θ(h))+θ(g)withg,h∈G θ :G→E isaone-cocycleofGwithvalues inE, q(st)= f(s)q(t)+q(s) a(X)(x)= r(X)(x)+Θ(X)withX∈ g,x∈E The linearmapΘ : g→E isaone-cocycleofGwith values inE:Θ(X)=Teθ(X(e)) ,X∈ g v → f(X)v+q(Y) f andq thedifferentialof fandq respectively Θ([X,Y])= r(X)(Θ(Y))−r(Y)(Θ(X)) q([X,Y])= f(X)q(Y)− f(Y)q(X)∀X,Y∈ g with f : g→ gl(E)andq : g →E none af f(X)= [ f(X) q(X) 0 0 ] none Af f(s)= [ f(s) q(s) 0 1 ] 6.8.AdditionalElementsonKoszulAffineRepresentationofLieGroupandLieAlgebra Let { x1,x2,...,xn } bealocalcoordinatesystemonM, theChristoffel’ssymbolsΓkijoftheconnection Daredefinedby: D ∂ ∂xi ∂ ∂xj = n ∑ k=1 Γkij ∂ ∂xk (112) ThetorsiontensorTofD isgivenby: T(X,Y)=DXY−DYX− [X,Y] (113) T ( ∂ ∂xi , ∂ ∂xj ) = n ∑ k=1 Tkij ∂ ∂xk withTkij=Γ k ij−Γkji (114) Thecurvature tensorRofD isgivenby: R(X,Y)Z=DXDYZ−DYDXZ−D[X,Y]Z (115) 77