Page - 152 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 (2) : [δτ f](ξ)= q+1 ∑ 1 (−1)i[ai · f(∂iξ)− f(ai ·∂iξ)+(f(∂2i,q+1ξ⊗ai)) ·aq+1] ∀f ∈Cqτ(A,W). Thepair (C∗τ(A,W),δτ) isacochaincomplex,viz δ2τ=0. Thederivedcohomologyspace isdenotedby Hτ(A,W)=∑ q Hqτ(A,W). It is calledtheW-valuedtotalKVcohomologyofA. 3.2. TheTheoryofKVCohomology—Version: theSemi-SimplicialObjects LetVbeatwo-sidedmoduleofaKValgebraA.Ouraimis theconstructionofsemisimplicial A-moduleswhosederivedcochaincomplex isquasi isomorphic to theKVcochincomplexCKV(A,V). 3.2.1. Extension Westartbyconsideringthevectorspace B=A⊕R. Itselementsaredenotedby (s+λ).WeendowBwiththemultiplicationwhich isdefinedby (s+λ) ·(s∗+λ∗)= s ·s∗+λs∗+λ∗s+λλ∗. With themultiplicationwe justdefined,B isa realKValgebra. Inotherwordswehave KV(X1,X2,X3)=0. Here Xj= sj+λj. In theA-moduleVwehaveastructureof leftB-modulewhich isdefinedby (s+λ)·v= s·v+λv ∀(s+λ)∈B, ∀v∈V. 3.2.2.Construction Let B˜ be the vector space spanned byA×R. Its elements are finite linear combinations of (s,λ),s∈A×R. Thetensoralgebraof B˜ isdenotedbyT(B˜). IthasaZ-grading. itshomogeneousvectorsub-spaces aredefinedby Tq(B˜)= B˜⊗q. Amonomialelement isdenotedby ξ= x1⊗x2⊗ ...⊗xq. Here xj=(sj,λj)∈A×R. 152