Page - 149 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 AvectorbundleV is a two-sidedKVmoduleofaKoszul-Vinbergalgebroid(E,b) if it satisfies the identities L(s, f,v)=0, KV(s,s∗,v)=0, KV(s,v,s∗)=0. Warning. ConsideravectorV spaceas the trivialvectorbundle V×O→0. Thenweget Γ(V×0)=V. Therefore analgebra is ananchoredalgebroidover apoint; its anchormapof is the zeromap. Therefore, theLeibnizanomalyofanalgebra isnothingbut thebilinearityof themultiplication. So thenotionofKValgebra andKVmodule is clear. 3.TheTheoryofCohomologyofKVAlgebroidsandTheirModules Thissection isdevotedto thecohomologyofKValgebroidsandKVmodulesofKValgebroids. KVstands forKoszul-Vinberg.Weshall introduce threeapproaches to the theoryofKVcohomology. Eachapproachhasitsparticularadvantage. So,dependingontheneedsorontheconcernsoneorother approachmaybeconvenient. The threeapproachesarecalled“Versionbrute formula”,“Versionsemi simplicial objects”, “Versionanomaly functions”. Thesamegradedvector space is commonto the threeconstructions. Theydiffer in theircoboundaryoperators.However, threeconstructions leadto cohomologycomplexeswhicharepairwisequasi isomorphic. Eachconstructionleadstotwocochaincomplexes.ThosecomplexesarecalledtheKVcomplexand totalKVcomplex.TheyaredenotedbyC∗KV andC∗τ. Infinalweobtainsixcohomologicalcomplexes. 3.1. TheTheoryofKVCohomology—Version theBruteFormulaof theCoboundaryOperator Thegeometric framework is thecategoryof realKValgebraoidsandtheir twosidedmodules. HoweverourmachineriesonlymakeuseofR-multi-linearcalculations in thevectorspacesof sections ofvectorbundles.Withoutanydamagewereplace thecategoriesofKValgebroidsandmodulesofKV algebroidsbythecategoriesofKValgebrasandabstractmodulesofKValgebras. 3.1.1. TheCochainComplexCKV. LetWbeatwo-sidedmoduleofaKValgebraA. Definition11. Thevector subspace J(W)⊂Wisdefinedby (a ·b) ·w−a ·(b ·w)=0 ∀a,b∈A Weconsider theZ-gradedvectorspace CKV(A,W)=∑ q CqKV(A,W). 149