Page - 153 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 TheKV algebraA is a two-sided ideal of the KV algebraB. Thereby, the vector space B˜ is canonicallya leftmoduleofA. Wedefinethenatural two-sidedactionofR in B˜bysetting λ ·(s∗,λ∗)=(λs,λλ∗), (s∗,λ∗) ·λ=(λs∗,λ∗λ). TherebyeveryvectorsubspaceTq(B˜) isa leftKVmoduleofB.Here the leftactionofB inTq(B˜) isdefined (s+λ) ·ξ= s ·ξ+λξ. Beforecontinuingwerecall the (extended)actionofA in tensorspaceTq(B˜), s ·(x1⊗x2⊗ ...⊗xq)= q ∑ j=1 x1⊗x2...⊗s ·xj⊗ ...⊗xq. Werecallanotationwhichhasbeenusedin the last subsections, ∂jξ= x1⊗x2⊗ ...xˆj...⊗xq. Thesymbol xˆjmeans thatxj ismissing. Let1∈Rbe theunitelement, then 1˜ stands for (0,1)∈B˜. Wearegoingtoconstruct semisimplicialmodulesofB. 3.2.3.Notation-Definitions Implicitlyweuseset isomorphism B˜ x=(s,λ)→X∗= s+λ∈B. Then∀ξ∈Tq(B˜)onehas 1˜∗ ·ξ= ξ· Wegobackto theZ-gradedB-module T∗(B˜)=∑ q Tq(B˜). Definition12. Let j,qbe twopositive integerswith j< q, let ξ= x1⊗x2...⊗xq. The linearmaps dj :Tq(B˜)→Tq−1(B˜) and Sj :Tq(B˜)→Tq+1(B˜) aredefinedby djξ=X∗j ·∂jξ, Sjξ= ej(1˜)ξ Therightmemberof the lastequalityhas the followingmeaning ej(1˜)ξ= x1⊗x2...⊗xj−1⊗ 1˜⊗xj...⊗xq 153