Page - 123 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 254 For instance, ifG is thegroupofEuclideantransformations,werecover theclassicalEuclidean tensors. Hence, each G-tensor can be identified with an orbit of G within the space of the tensorcomponents. 3.MomentumasAffineTensor LetM be a differential manifold of dimension n and G a Lie subgroup ofAf f(n). In the applications tophysics,Mwillbe forus typically thespace-timeandGasubgroupofAf f(n)witha physicalmeaning intheframeworkofclassicalmechanics (Galileo’sgroup)orrelativity (Poincaré’s group). Thepointsof thespace-timeMareeventsofwhichthecoordinateX0 is thetime tandXi= xi for i runningfrom1to3gives theposition. The tangent space toM at thepointX equippedwitha structureof affine space is called the affine tangent spaceand isdenotedATXM. Its elementsare called tangentpoints atX. The setof affineformsontheaffinetangentspace isdenotedA∗TXM.Wecallmomentumabilinearmapμ: μ :TXM×A∗TXM→R : (−→V ,Ψ) →μ(−→V ,Ψ) It isamixed1-covariantand1-contravariantaffine tensor. Taking intoaccount thebilinearity, it is represented inanaffineframe f by: μ( −→ V ,Ψ)=(χKβ+ΦαLαβ)V β whereKβ andLαβ are thecomponentsofμ in theaffineframe f or,equivalently, thecoupleμ=(K,L) of therowK collecting theKβ andthen×nmatrixLofelementsLαβ. Owingto (2), the transformation lawisgivenbythe inducedactionofAf f(n): K′=KP−1, L′=(PL+CK)P−1 (3) If theaction is restrictedto thesubgroupG, themomentumμ isaG-tensor. Ontheotherhand,havea lookto theLiealgebragofG, that is thesetof infinitesimalgenerators Z= da=(dC,dP)with a∈G. Letus identify thespaceof themomentumcomponentsμ=(K,L) to thedualg∗of theLiealgebra thanks to thedualpairing: μZ=μda=(K,L)(dC,dP)=KdC+Tr(LdP) (4) Weknowthat thegroupactson itsLiealgebrabytheadjoint representation: Ad(a) :g→g :Z′ →Z=Ad(a)Z′= aZ′a−1 . AsG isagroupofaffinetransformations,any infinitesimalgeneratorZ is representedby: Z˜= dP˜= d ( 1 0 C P ) = ( 0 0 dC dP ) . Then Z˜= P˜ Z˜′ P˜−1 leads to: dC=P(dC′−dP′P−1C), dP=PdP′P−1 . (5) Thisadjoint representation induces thecoadjoint representationofG ing∗definedby: (Ad∗(a)μ′)Z=μ′(Ad(a−1)Z) . 123