Abstract:
Let $K$ be a base field of characteristic zero. It is well known that in this case all information about varieties of linear algebras $\bf{V}$ contains in its polylinear components $P_n(\bf{V})$, $n \in \mathbb{N}$, where $P_n(\bf{V})$ is a linear span of polylinear words of $n$ different letters in a free algebra $K(X,\bf{V})$.
D. Farkas defined customary polynomials and proved that every Poisson PI-algebra satisfies some customary identity.
Poisson algebras are special case of Leibniz–Poisson algebras.
In the paper the sequence of customary spaces of the free Leibniz–Poisson algebra $\{Q_{2n}\}_{n\geq 1}$ is investigated. The basis and dimension of spaces $Q_ {2n}$ are given.
It is also proved that in case of a base field of characteristic zero any nontrivial identity of the free Leibniz–Poisson algebra has nontrivial identities in customary spaces.
Key words:Poisson algebra, Leibnitz–Poisson algebra, variety of algebras, growth of variety.