On Leibniz–Poisson Special Polynomial Identities

In this paper we study Leibniz–Poisson algebras satisfying polynomial identities. We study Leibniz–Poisson special and Leibniz–Poisson extended special polynomials. We show that the sequence of codimensions $\{r_n({\mathbf V})\}_{n\geq 1}$ of every extended special space of variety ${\mathbf V}$ of Leibniz-Poisson algebras over an arbitrary field is either bounded by a polynomial or at least exponential. Furthermore, if this sequence is bounded by polynomial then there is a polynomial $R(x)$ with rational coefficients such that $r_n({\mathbf V}) = R(n)$ for all sufficiently large n. It follows that there exists no variety of Leibniz-Poisson algebras with intermediate growth of the sequence $\{r_n({\mathbf V})\}_{n\geq 1}$ between polynomial and exponential. We present lower and upper bounds for the polynomials $R(x)$ of an arbitrary fixed degree.