Page - 158 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 SomeComments. (c.1): Wereplace theKVB byA. Thenweobtain the exact sequences →Hq−1KVres(A,V)→Hqe(A,V)→HqKV(A,V)→HqKVres(A,V)→ →Hq−1τres(A,V)→Hqe(A,V)→Hqτ(A,V)→Hqτres(A,V)→ (c.2): The KV cohomology difers from the total cohomology. Loosely speaking their intersecttion is the equivariant cohomology H∗e(B,V) their difference is the residual cohomology. The domain of their efficiencyaredifferentaswell.Hereare two illustrations. Example1. In the introductionwehavestatedaconjectureofM.Gerstenhaber,namelyEveryRestrictedTheory ofDeformationGenerates ItsProperTheoryofCohomology. From the viewpoint of this conjecture, the KV cohomology is the completion a long history [2,9,28]. BesidesKoszul andNijenhuis, otherpioneering authors areVinberg, Richardson, Gerstenhaber,Matsushima,Vey. Thechallengewas thesearch fora theoryofcohomologywhichmightbegeneratedbythe theory ofdeformationof locallyflatmanifolds [8]. Theexpectedtheory is thenowknownKVtheoryofKV cohomolgy[9]. Example2. The total cohomology is close toboth thepioneeringNijenhuiswork [28,36]. In [29]wehave constructedaspectral sequencewhichrelates to [28,36]. From another viewpoint, the total KV cohomology is useful for exploring the relationships between the informationgeometryand the theoryofRiemannian foliations. Thispurposewill be addressed in thenextsections. 3.3. TheTheoryofKVCohomology—Version theAnomalyFunctions This subsection is devoted to use the KV anomaly functions for introducing the theory of cohomologyofKValgebroidsandtheirmodules. This viewpoint leads to anunifying framework for introducing the theoryof cohomologyof abstractalgebrasandtheirabstract two-sidedmodules.Hereareafewexamplesofcohomologytheory whicharebasedontheanomalyfunctions. Example1.The theoryofHochschildcohomologyofassociativealgebras isbasedontheassociator anomalyfunction. Example2.ThetheoryofChevalley-Eilenberg-KoszulcohomologyofLiealgebras isbasedontheJacobi anomalyfunction. Example 3. The theoryof cohomologyofLeibniz algebras is basedon the Jacobi anomaly function aswell. 3.3.1. TheGeneralChallengeCH(D) Weconsiderdata D=[(A,AA),(V,AAV),Hom(T(A),V)]. Here (1) V isan(abstract) twosidedmoduleofan(abstract)algebraA. (2) AA andAAV arefixedanomalyfunctionsofAandofV respectively. 158