Page - 161 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 Ouraimis toevaluateξofE∗∗∗∗(f)atξ∈A⊗q+2.Here ξ=X1⊗ ...⊗Xq+2. Tocalculate [E∗∗∗∗(f)](ξ)wetake intoaccountbothSTEP1andSTEP2. Thenweobtain [E∗∗∗∗(f)](ξ)= ∑ [i<j<q+2;1≤k≤q+2] [E∗∗∗∗[ijk] (f)](ξ). At therightsidemember [E∗∗∗∗[ijk] (f)](ξ)=(−1)i+j[KV(Xi,Xj, f(X1⊗ ...Xˆi⊗ ..Xˆj⊗ ...Xk⊗ ...⊗Xq+2)) +KV(Xi, f(X1⊗ ...Xˆi...Xˆj⊗ ...⊗Xq+1⊗Xj),Xq+2) +KV(Xj, f(X1⊗ ...Xˆi...Xˆj⊗ ...⊗Xq+1⊗Xi),Xq+2) +ω(f)(ω(f)+1)f(X1⊗ ...Xˆi...Xˆj⊗ ...⊗KV(Xi,Xj,Xk)⊗ ...⊗Xq+2)]. Step4. Weare inpositionto faceCH(D). Definition 17. Let f ∈Hom(A⊗q,W) and ξ =X1⊗ ...⊗Xq+1 ∈A⊗q+1. We take into account Step 1, Step2andStep3. Therefore,wedefine the linearmap Hom(A⊗q,W) f→∂f ∈Hom(A⊗q+1,W) byputting [∂f](ξ)= ∑ 1≤i<j≤q+1 S[i,j](f)(ξ) Thefollowing lemmaisastraightforwardconsequenceof themachinery inSTEP3. Lemma2. ∂2 f(ξ)= ∑ [i<j<q+2];1≤k≤q+2 [E∗∗∗∗[ijk] (f)](ξ) Lemma2tellsus that∂2 f(ξ)depends linearlyonthevaluesof theKVanomalyfunctions. ThechallengeCH(D) iswoninthecategoryofKValgebrasandtheir two-sidedKVmodules. We replace the category ofKValgebras and their two-sidedmodules by the category ofKV algebroidsandtheirbi-modules. ThenwewinthegeometryversionofCH(D. We use Lemma 2 for introducing a theory of KV homology of KV algebras and their two-sidedmodules. 3.3.3. TheKVCohomology LetWbeatwosidedKVmoduleofaKValgebraA.Weconsider thegradedvectorspace CKV=⊕qCqKV. The homogeneous subspaces are definedby CqKV = 0 if q is a negative integer, C0KV = J(W), CqKV=Hom(A⊗q,W) ifq isapositive integer. Wedefinethe linearmap CqKV f→∂KVf ∈Cq+1KV 161