Page - 179 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 Ci,j=Ciτ(A∗,A∗)⊗CjKV(A,R). Werecall thatC∗(A,R)stands forC∗(A,C∞(M)). Foreverynonnegative integerqweset Cq=Σi+j=qCi,j. Wedefines the linearmap δi,j :Ci,j→Ci+1,j⊕Ci,j+1 by δi,j= δτ⊗1+(−1)i⊗δτ. Soweobtaina linearmap Cq→Cq+1 Therefore,weconsider thebi-gradeddifferentialvectorspace C :=(C∗∗,δ∗∗). That isabi-gradedcochaincomplexwhoseqthcohomology isdenotedbyHq(C). Thecohomology inherits thebi-grading Hq(C)= ∑ [i+j=q] Hi,j(C). Here Hi,j(C)= Ci,j∩ [Ziτ(A∗,A∗)⊗Zjτ(A,R)] im(δi−1,j)+ im(δi, j−1) In thenextsubsubsectionweshalldiscuss the impactsof thiscohomology. Remark2. Thepair (C∗∗,δ∗∗)generates a spectral sequence [34]. That spectral sequence is auseful tool for simultaneously computing both theKVcohomology and the total KV cohomology ofKValgebroids. Those matters arenot thepurposeof thispaper. 5.1.3. TheHomologicalNatureofGaugeHomomorphisms Givingaduallyflatpair (M,g,D,D∗)oneconsiders the linearmap C1,0τ (A∗,A∗) ψ→ψ⊗qψ∈C1,2. Here thesymmetric2-formqψ isdefinedby qψ(X,Y)= 1 2 [g(ψ(X),Y)+g(X,ψ(Y))]. To relate the bi-complex (C∗∗,δ∗∗) and the space of gauge homomorphisms we use the followingstatement. Theorem9. Givenagaugemorphism ψ :TM→TM the followingstatementsare equivalent (1) ψ∈M(D,D∗), (2) δ1,2(ψ⊗qψ)=0 179