Page - 154 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 Structure.Themapsdj andSj satisfy the following identities didj=dj−1di if i≤ j, (5a) SiSj=Sj+1Si if i< j, (5b) (Sj−1di−diSj)(ξ)= ej−1(xi)∂iξ if 1< i< j, (5c) (di+1Sj−Sjdi)(ξ)= ej(xi)∂iξ if j+1< i≤ q, (5d) di(Si(ξ))= ξ if i= j. (5e) Definition13. Thesystem { Tq(B),di,Si } is called the canonical semi simplicialmoduleofB. 3.2.4. TheKVChainComplex Fromthecanonical simplicialB-modulewederive thechaincomplexC∗(B). ithasaZ-grading which isdefinedby Cq(B)=0 if q<0, (6a) C0(B)=R, (6b) Cq(B)=Tq(B˜) if q>0. (6c) Nowonedefines the ( linear)boundaryoperator d :Cq(B)→Cq−1(B) bysetting d(C0(B))=0, d(C1(B))=0, dξ= q ∑ 1 (−1)jdjξ if q>1. Bythevirtueof (5a)wehave d2=0. 3.2.5. TheV-ValuedKVHomology Wekeepthenotationusedin theprecedingsub-subsection. So thevectorspacesA,BandVare thesameas in theprecedingsubsubsection. Weconsider theZ-gradedvectorspace C∗(B,V)=⊕qCq(B,V). Itshomogeneoussub-spacesaredefinedby Cq(B,V)=0 if q<0, C0(B,V)=V, Cq(B,V)=Tq(B˜)⊗V if q>0. EveryhomogeneousvectorsubspaceCq(B,V) isa leftmoduleof theKValgebraB. The leftaction isdefinedby s ·(ξ⊗v)= s ·ξ⊗v+ξ⊗s ·v. 154