Page - 160 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 +(−1)i[Xi · f(X1⊗ ...Xˆi⊗ ...⊗Xj⊗ .⊗Xq+1) +(f(X1⊗ ...Xˆi⊗ ...⊗Xj⊗ ... ˆXq+1⊗Xi)) ·Xq+1 −ω(f)f(X1⊗ ...Xˆi⊗ ...⊗Xi ·Xj⊗ ...⊗Xq+1)]. In therightsidememberofS[i,j](f)(ξ) thecoefficientω(f) is thedegreeof f, vizω(f)= q forall f ∈Hom(Aq,W). Step2. Foreverypair (i,q+1)with1≤ i≤ qwedefinethemapS[i,q+1](f)by S[i,q+1](f)(X1⊗ ...⊗Xq+1)=(−1)i[Xi · f(X1⊗ ...Xˆi⊗ ...⊗Xq+1) +(f(X1⊗ ...Xˆi⊗ ... ˆXq+1⊗Xi)) ·Xq+1 −ω(f)f(X1⊗ ...Xˆi⊗ ...Xi ·Xq+1)]. Step3. Letg∈Hom(A⊗q+1,W)andlet ξ=X1⊗ ...⊗Xq+2∈A⊗q+2. Let i, j,kbethreepositiveintegerssuchthat i< j< q+2; k≤ q+2.Wehavealreadyintroduced thenotation ∂kξ=X1⊗ ...Xˆk⊗ ...⊗Xq+2, ∂2k,q+2ξ=X1⊗ ...Xˆk⊗ ...⊗ ...Xˆq+2. WedefineSk [i,j](g)∈Hom(A⊗q+2,W)bysetting Sk[i,j](g)(ξ)=(−1)i+k[Xk ·g(∂kξ)+(g(∂2k,q+2ξ⊗Xk)) ·Xq+2 +ω(g)g(X1⊗ ...⊗Xk ·Xi⊗ ...Xˆk⊗ ...⊗Xq+2)] +(−1)j+k[Xk ·g(∂kξ)+(g(∂2k,q+2ξ)⊗Xk) ·Xq+2 +ω(g)g(X1⊗ ...⊗Xk ·Xj⊗ ...Xˆk⊗ ...⊗Xq+2)]. Givenatriple (i, j,k)with i< j< k< q+2weput S[i,j,k](g)(ξ)=S k [i,j](g)(ξ)+S j [i,k](g)(ξ)+S i [j,k](g)(ξ). Theproofof the followingstatement isbasedondirect calculations. Lemma1. (∗∗∗∗) : ∑ [i<j] S[i,j](g)(ξ)= ∑ [i<j<k] S[i,j,k](g)(ξ) Let f ∈Hom(Aq,W). Inboththe left sideandtherightsideof theequality (∗∗∗∗)wereplaceg by∑i<jS[i,j](f). Thenweobtaina linearmapping Hom(A ,W) f→E∗∗∗∗(f)∈Hom(Aq+2,W). 160