Page - 151 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 (2)LetWbea two-sidedmoduleof aKValgebraA.WeconsiderWasa trivialKValgebra,viz w ·w∗=0 ∀w,w∗∈W. LetEXTKV(A,W)be the set of equivalence classes of short exact sequencesofKValgebras 0→W→B→A→0. Aninterpretationof the2ndcohomologyspaceofCKV(A,W) is the identification H2KV(A,W)=EXTKV(A,W). LetW,W∗be two-sidedmodulesofA. LetEXTA(W∗,W)be the set of equivalence classesof exact short sequencesof two-sidedA-modules 0→W→T→W∗→0. Inboth the categoryof associativealgebrasand the categoryofLie algebraswehave HH1(A,HomR(W∗,W))=EXTA(W∗,W), H1CE(A,HomR(W∗,W))=EXTA(W∗,W). HereHH(A,−) stands forHochschild cohomologyof anassociativealgebraAandHCE(A,−) stands for cohomologyofChevalley-Eilenbergof aLie algebraA. Unfortunately in the categoryofKVmodules ofKValgebras this interpretationof thefirst cohomology space fails. Loosely speaking in the category ofKValgebras the setH1(A,Hom(W∗,W)) is not canonically isomorphic to setEXTA(W∗,W) [9]. 3.1.2. TheTotalCochainComplexCτ. Thepurpose is the total complex Cτ(A,W)=∑ q Cqτ(A,W). Itshomogeneousvectorsubspacesaredefinedby Cqτ(A,W)=0 ∀q<0, C0τ(A,W)=W, Cqτ(A,W)=HomR(A⊗q,W) ∀q>0. Thetotal coboundaryoperator isa linearmap Cqτ(A,W)→Cq+1τ (A,W). Thatoperator isdefinedby (1) : [δτw](a)=−a ·w+wa ∀(a,w)∈A×W, 151