Poisson-even Algebras And Realizations Of3-Lie Algebras

In recent years,3-Lie algebras are applied to many branches of mathematics andmathematical physics. In the paper we mainly study the realizations of3-Lie algebras.Inspired by Poisson algebras, we provide a new algebra which is called Poisson-evenalgebra, and construct3-Lie algebras basing on it.Poisson-even algebra (L,,[,]) is a commutative associative algebras (L,) with Lieproduct [,], and two binary operations and [,] satisfy [z x, y]+[x, z y]=2z [x, y]for any x, y, z∈L. It has close relationship with Hom-Lie algebras and Left-Poissonalgebras. Suppose f∈End(L), and define a ternary operation [,,]: L×L×Lâ†'L,[x, y, z]=f (x)[y, z]+f (y)[z, x]+f (z)[x, y] for any x, y, z∈L. We prove that iff is an involution or derivation,(L,[,,]) is a3-Lie algebra. Finally, we construct3-Liealgebras on Laurent polynomials algebras, and discuss its structure.