Abstract
In this paper, based on the existing research results, we obtain the unary extension 3-Lie algebras by one-dimensional extension of the known Lie algebra L. For two known 3-Lie algebras H, M, the (μ, ρ, β)-extension of H through M is given, and the necessary and sufficient conditions for the (μ, ρ, β)-extension algebra of H through M being 3-Lie algebra are obtained, and the structural characteristics and properties of these two kinds of extended 3-Lie algebras are given.
Highlights
The study of 3-Lie algebra has been paid much attention because of its wide application in mathematics and physics. 3-Lie algebra is a special form of n-Lie algebra, which is an algebraic system with ternary linearly oblique symmetric multiplication table satisfying the generalized Jacobi equation [1]. 3-Lie algebra has extremely profound and rich algebraic and analytical structure
The extension problem of 3-Lie algebra is studied on the basis of the existing research
The structure and properties of this extended 3-Lie algebra are discussed. It lays a foundation for the further study of the properties of the derivatives of two kinds of 3-Lie algebras
Summary
The study of 3-Lie algebra has been paid much attention because of its wide application in mathematics and physics. 3-Lie algebra is a special form of n-Lie algebra, which is an algebraic system with ternary linearly oblique symmetric multiplication table satisfying the generalized Jacobi equation [1]. 3-Lie algebra has extremely profound and rich algebraic and analytical structure. The extension problem of 3-Lie algebra is studied on the basis of the existing research. We define the unary extended 3-Lie algebra for a known Lie algebra L by one-dimensional extension, and study its properties. For two known 3-Lie algebras H, M, the (μ, ρ, β ) -extension of H through M is defined, and the (μ, ρ, β ) -extension of H through M is given as a necessary and sufficient condition for the 3-Lie algebra. The structure and properties of this extended 3-Lie algebra are discussed. It lays a foundation for the further study of the properties of the derivatives of two kinds of 3-Lie algebras
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