Page - 159 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 (3) Hom(T(A),V)stands for theZ-gradedvectorspace Hom(T(A),V)=⊕qHomR(A⊗q,V). LetADbe thecategoryof (abstract)algebrasand(abstract)moduleswhosestructuresaredefined bythepair (AA,AAV). So therulesofcalculations in thecategoryAaredefinedbythe identities AA(a,b,c)=0, AAV(a,b,v)=0. Thechallenge is thesearchofaparticular familyof linearmaps Hom(A⊗q,V) f→dq(f)∈Hom(A⊗q+1,V). Suchaparticular familydqmustsatisfyaconditionthatwecall thepropertyΔ. PropertyΔ ∀ξ= a1⊗a2...⊗aq+2∈Aq+2,∀f ∈Hom(A⊗q,V) thequantity [dq+1(dq(f))](ξ)depends linearlyon thevaluesof theanomaly functions { AA(ai,aj,ak), AAV(ai,aj,v) } Letusassumethata familydq isasolutiontoCH(D). ThenthecategoryADadmitsa theoryof cohomologywithcoefficients inmodules. The next is devoted to this challenge in the category of KV algebras and KV modules. Thegeometryversion is thecategoryofKValgebroidsandKVmodulesofKValgebroids. 3.3.2.ChallengeCH(D) forKVAlgebras LetWbea two-sidedmoduleofanabstract algebraA. Weassumethat the followingbilinear mappingsarenontrivialapplications A×W (X,w)→X ·w∈W, W×A (w,X)→w ·X∈W. Let f ∈Hom(A⊗q,W).Weconsideramonomialξ∈A⊗q+1, so ξ=X1⊗ ...⊗Xq+1∈A⊗q+1. Ourconstruction isdividedintomanySTEPS. Step1. Let (i< j)beapairofpositive integerswith1≤ i< j≤ q. The linear themap S[i,j](f)∈Hom(Aq+1,V). S[ij] isdefinedby S[i,j](f)(X1⊗ ...⊗Xq+1)=(−1)j[Xj · f(X1⊗ ...⊗Xi⊗ ...Xˆj⊗Xj+1...⊗Xq+1) +(f(X1⊗ ...⊗Xi⊗ ...Xˆj⊗ ... ˆXq+1⊗Xj) ·Xq+1 −ω(f)f(X1⊗ ...⊗Xj ·Xi⊗ ...Xˆj⊗ ...⊗Xq+1)] 159