Page - 75 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 386 AHessian structure (D, g) on a homogeneous spaceG/K is said to be an invariantHessian structure if bothD and g areG-invariant. A homogeneous spaceG/Kwith an invariantHessian structure (D, g) is calledahomogeneousHessianmanifoldand isdenotedby (G/K,D,g). Another result ofKoszul is that ahomogeneous self-dual regular convex cone is characterizedas a simply connectedsymmetrichomogeneousspaceadmittingan invariantHessianstructure that isdefined bythepositivedefinitesecondKoszul form(wehave identifiedinapreviouspaper that this second Koszul form is related to the Fishermetric). In parallel, Vinberg [125,126] gave a realization of a homogeneous regular convex domain as a real Siegel domain. Koszul has observed that regular convexconesadmitcanonicalHessianstructures, improvingsomeresultsofPyateckii-Shapiro that studiedrealizationsofhomogeneousboundeddomainsbyconsideringSiegeldomains inconnection with automorphic forms. Koszuldefineda characteristic functionψΩ of a regular convex coneΩ, andshowedthatψΩ=DdlogψΩ isaHessianmetriconΩ invariantunderaffineautomorphismsofΩ. IfΩ isahomogeneousselfdualcone, thenthegradientmapping isasymmetrywithrespect to the canonicalHessianmetric, andisasymmetrichomogeneousRiemannianmanifold.More information onKoszulHessiangeometrycanbefoundin[127–136]. Wewillnowfocusourattention toKoszulaffinerepresentationofLiegroup/algebra. LetGa connexLiegroupandEarealorcomplexvectorspaceoffinitedimension,Koszulhas introducedan affinerepresentationofG inE suchthat [117–124]: E→E a → sa∀s∈G (97) isanaffinetransformation.WesetA(E) thesetofall affinetransformationsofavectorspaceE, aLie groupcalledaffine transformationgroupofE. ThesetGL(E)ofall regular linear transformationsofE, a subgroupofA(E). Wedefinea linearrepresentationfromG toGL(E): f :G→GL(E) s → f(s)a= sa−so∀a∈E (98) andanapplicationfromG toE: q :G→E s →q(s)= so∀s∈G (99) Thenwehave∀s,t∈G: f(s)q(t)+q(s)=q(st) (100) deducedfrom f(s)q(t)+q(s)= sq(t)−so+so= sq(t)= sto=q(st). On thecontrary, if anapplication q fromG toEanda linear representation f fromG toGL(E) verifypreviousequation, thenwecandefineanaffinerepresentationofG inE,written (f,q): Af f(s) : a → sa= f(s)a+q(s)∀s∈G,∀a∈E (101) The condition f(s)q(t)+q(s) = q(st) is equivalent to requiring the followingmapping tobe anhomomorphism: Af f : s∈G →Af f(s)∈A(E) (102) 75