Page - 150 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 Thehomogeneousvectorsub-spacesaredefinedby CqKV(A,W)=0 ∀q<0, C0KV(A,W)= J(W), CqKV(A,W)=HomR(A⊗q,W) ∀q>0. Beforepursuingwefixthe followingnotation. Let ξ= a1⊗ ...⊗aq+1∈A⊗q+1 andlet a∈A, ∂iξ= a1⊗ ...aˆi...⊗aq+1, ∂2i,k+1ξ=∂i(∂k+1ξ), a.ξ= q+1 ∑ 1 a1⊗ ...aj−1⊗a.aj⊗aj+1...aq+1. Wearegoingtodefinethecoboundaryoperator δKV :Cq(A,W)→Cq+1(A,W). Thecoboundaryoperator isa linearmap. It isdefinedby [δKV(w)](a)=−a ·w+w ·a ∀w∈ J(W), (4a) [δKVf](ξ)= q ∑ 1 (−1)i[ai · f(∂iξ)− f(ai ·∂iξ)+(f(∂2i,q+1ξ⊗ai)) ·aq+1]∀f ∈CqKV(A,W), ∀ξ∈A⊗q+1. (4b) TheoperatorδKV satisfies the identity δ2KVf=0 ∀f ∈CKV(A,W). Therefore thepair (C∗KV(A,W),δKV) isacochaincomplex. Its cohomologyspace isdenotedby HKV(A,W)=∑ q HqKV(A,W). TheconjectureofGerstenhaber:Comments. AKValgebraA is a two-sidedmodule of itself. An infinitesimal deformations ofA is a 1-cocycle of CKV(A,A) [9]. By the conjecture ofGerstenhaber the cohomology complexCKV(A,A) is generated by the theoryofdeformations in the categoryofKValgebras. The theoryofdeformationofKValgebras is thealgebraicversionof the theoryofdeformationof locallyflat manifolds [2]. Therefore, the complexCKV(A, ) is the solution to the conjectureofMurayGerstenhaber in the categoryof locallyflatmanifolds [27]. Features. (1)The2ndcohomologyspaceH2KV(A,A) is the spaceofnontrivialdeformationsofA. ThedefinitionofKValgebraof a locallyflatmanifoldwill begiven in thenext section. Following [2] everyhyperbolic locallyflatmanifoldhasnon trivial deformations. Thus, ifA is theKValgebraof ahyperbolic locallyflatmanifold then H2KV(A,A) =0. 150