On the nilpotent Leibniz–Poisson algebras

In this article Leibniz and Leibniz–Poisson algebras in terms of correctness of different identities are investigated. We also examine varieties of these algebras. Let $K$ be a base field of characteristics zero. It is well known that in this case all information about varieties of linear algebras $V$ contains in its polylinear components $P_n(V)$, $n \in \mathbb{N}$, where $P_n(V)$ is a linear span of polylinear words of $n$ different letters in a free algebra $K(X,V)$. In this article we give algebra constructions that generate class of nilpotent varieties of Leibniz algebras and also algebra constructions that generate class of nilpotent by Leibniz varieties of Leibniz–Poisson algebras with the identity $\{ x_1, x_2 \} \cdot \{x_3, x_4 \} = 0$.