Left On The Mold Lie Algebra W (m, N, (?)) Supersymmetric Structure

This paper discusses some properties about etale superaffine rep-resentation by defining left-supersymmetric algebra .It discusses the re-lationship between left-supersymmetric structures on Lie superalgebra and it's 1't-h cohomology group. Let W := W(m,n,t) be the gen-eral module Lie superalgebra of Cartan type. Under certain conditions, translative isomorphisms induce left-supersymmetric structures on W. On the other hand, the mixed product theorems are generalized.The main results in this paper are the following: Theoreml: Let L be the extension of a subalgebra L0 of Lie superalgebra gl(s, F) with respect to a displacement ψ p a representation of Lo in the Z2-space V. And ~p the representation of rj(g)L0 in Z2- graded space 17 V extending p. Then a :A-盇 lv+~p(p(A)) is a representation of L in the 2- graded space 17 V.Theorem2: LetL = L0 0 L\ be an n-dimensional Lie super-algebra which satisfies [L,L] = L with dimL0 dimLi.lf for any L- module structure on V = L.we have Hl(L,V) = O.Then there is no left-supersymmetric structure on L.TheoremS: Let the ground field F be arbitrary of characteristic p > 3. Then 1) W adimits left-supersymmetric structures; 2) If F is infinite, then W admits infinitely many left-supersymmetric structures.