

Flexible Lieadmissible Superalgebras of Vector Type 
G. Lakshmi Devi and K. Jayalakshmi 
Abstract:
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 δ_{x,y} : a ↦ (a, x, y) the derivation D : a ↦ (x, a, x) and the derivation D_{ij} : a ↦ (x_{i} ,a, x_{j}) Results: The flexible Lieadmissible superalgebra F_{FLSA}[φ; x] over a 2, 3torsion free field Φ on one odd generator e is isomorphic to the twisted superalgebra B_{0} (Φ[Γ], D, γ_{0}) with the free generator . In a 2, 3torsion free flexible Lieadmissible 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 Abimodule. Conclusion: A connection between the integral domains, the finitely generated projective modules over them, the derivations of an integral domain and the flexible Lieadmissible superalgebras of vector type has been established. If A is an integral domain and M = Ax_{1}+…+Ax_{n} be a finitely generated projective Amodule of rank 1, then F (A, Δ, Γ) is a flexible Lieadmissible superalgebra with even part A and odd part M provided that the mapping M = Ax_{i}+…+Ax_{n} is a nonzero derivation of A into the Amodule (M⊗_{A} M)*, Δ = {D_{ij} i, j = 1,…, n} is a set of derivations of A where D_{ij} (a) = ā (x⊗x_{j}).


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How to cite this article:
G. Lakshmi Devi and K. Jayalakshmi, 2017. Flexible Lieadmissible Superalgebras of Vector Type. Asian Journal of Algebra, 10: 19. DOI: 10.3923/aja.2017.1.9 URL: https://scialert.net/abstract/?doi=aja.2017.1.9




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