Page - 76 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 386 Wewrite f the linear representationofLiealgebragofG,definedby fandq therestriction togof thedifferential toq (f andq thedifferentialof fandq respectively),Koszulhasprovedthat: f(X)q(Y)− f(Y)q(X)= q([X,Y]) ∀X,Y∈ g with f : g→ gl(E)andq : g →E (103) wheregl(E) thesetofall linearendomorphismsofE, theLiealgebraofGL(E). Usingthecomputation, q(AdsY)= dq(s ·etY ·s−1) dt ∣∣∣∣ t=0 = f(s)f(Y)q(s−1)+f(s)q(Y) (104) Wecanobtain: q([X,Y])= dq(AdetXY) dt ∣∣∣∣ t=0 = f(X)q(Y)q(e)+f(e)f(Y)(−q(X))+ f(X)q(Y) (105) where e is theunitelement inG. Since f(e) is the identitymappingandq(e)=0,wehave theequality: f(X)q(Y)− f(Y)q(X)= q([X,Y]) . Apair (f,q)ofa linearrepresentation f ofaLiealgebragonEanda linearmappingq fromg toE isanaffinerepresentationofgonE, if it satisfies f(X)q(Y)− f(Y)q(X)= q([X,Y]) . Conversely, if we assume that g admits an affine representation (f,q) on E, using an affine coordinate system { x1,...,xn } on E, we can express an affinemapping v → f(X)v+q(Y) by an (n+1)×(n+1)matrixrepresentation: af f(X)= [ f(X) q(X) 0 0 ] (106) where f(X) isan×nmatrixandq(X) isan rowvector. X → af f(X) isan injectiveLiealgebrahomomorphismfromg in theLiealgebraofall (n+1)× (n+1)matrices,gl(n+1,R): ∣∣∣∣∣ g→ gl(n+1,R)X → af f(X) (107) Ifwedenotegaf f = af f(g),wewriteGaf f the linearLiesubgroupofGL(n+1,R)generatedby gaf f . Anelementof s∈Gaf f isexpressedby: Af f(s)= [ f(s) q(s) 0 1 ] (108) LetMaf f be the orbit ofGaf f through the origin o, thenMaf f = q(Gaf f) = Gaf f/Kaf f where Kaf f = { s∈Gaf f/q(s)=0 } =Ker(q). Example.LetΩbeaconvexdomain inRn containingnocompletestraight lines,wedefineaconvex coneV(Ω) inRn+1 = Rn×RbyV(Ω) = {(λx,x)∈Rn×R/x∈Ω,λ∈R+}. Then there exists an affineembedding: : x∈Ω → [ x 1 ] ∈V(Ω) (109) 76