Flexible Lie-admissible Superalgebras of Vector Type

摘要

Background: First examples of simple nonassociative superalgebras were constructed by Shestakov in (1991 and 1992). Since then many researchers showed interest towards the study of superalgebras and superalgebras of vector type. Materials and Methods: Multiplication in M is uniquely defined by a fixed finite set of derivations and by elements of A. The types of derivations used in this article to obtain the results are the near derivation δx,y : a ? (a, x, y) the derivation D : a ? (x, a, x) and the derivation Dij : a ? (xi ,a, xj) Results: The flexible Lie-admissible superalgebra FFLSA[φ; x] over a 2, 3-torsion free field Φ on one odd generator e is isomorphic to the twisted superalgebra B0 (Φ[Γ], D, γ0) with the free generator . In a 2, 3-torsion free flexible Lie-admissible superalgebras of vector type F, the even part A is differentiably simple, associative and commutative algebra and the odd part M is a finitely generated associative and commutative A-bimodule. Conclusion: A connection between the integral domains, the finitely generated projective modules over them, the derivations of an integral domain and the flexible Lie-admissible superalgebras of vector type has been established. If A is an integral domain and M = Ax1+…+Axn be a finitely generated projective A-module of rank 1, then F (A, Δ, Γ) is a flexible Lie-admissible superalgebra with even part A and odd part M provided that the mapping M = Axi+…+Axn is a nonzero derivation of A into the A-module (M?A M)*, Δ = {Dij |i, j = 1,…, n} is a set of derivations of A where Dij (a) = ā (x?xj).