Numerical characteristics of Leibniz–Poisson algebras

The paper is survey of recent results of investigations on varieties of Leibniz–Poisson algebras. We show that a variety of Leibniz–Poisson algebras has either polynomial growth or growth with exponential not less than 2, the field being arbitrary. We show that every variety of Leibniz–Poisson algebras of polynomial growth over a field of characteristic zero has a finite basis for its polynomial identities. We construct a variety of Leibniz–Poisson algebras with almost polynomial growth. We give equivalent conditions of the polynomial codimension growth of a variety of Leibniz–Poisson algebras over a field of characteristic zero. We show all varieties of Leibniz–Poisson algebras with almost polynomial growth in one class of varieties. We study varieties of Leibniz–Poisson algebras, whose ideals of identities contain the identity $\{x,y\}\cdot \{z,t\}=0$, we study an interrelation between such varieties and varieties of Leibniz algebras. We show that from any Leibniz algebra $L$ one can construct the Leibniz–Poisson algebra $A$ and the properties of $L$ are close to the properties of $A$. We show that if the ideal of identities of a Leibniz–Poisson variety $\mathbf{ V}$ does not contain any Leibniz polynomial identity then $\mathbf{ V}$ has overexponential growth of the codimensions. We construct a variety of Leibniz–Poisson algebras with almost exponential growth. Let $\{\gamma_n(\mathbf{ V})\}_{n\geq 1}$ be the sequence of proper codimension growth of a variety of Leibniz–Poisson algebras $\mathbf{ V}$. We give one class of minimal varieties of Leibniz–Poisson algebras of polynomial growth of the sequence $\{\gamma_n(\mathbf{ V})\}_{n\geq 1}$, i.e. the sequence of proper codimensions of any such variety grows as a polynomial of some degree $k$, but the sequence of proper codimensions of any proper subvariety grows as a polynomial of degree strictly less than $k$.
This work is devoted to the seventieth Doctor of Physical and Mathematical Sciences, Professor Vasily Ivanovich Bernik. In her curriculum vitae, a brief analysis of his scientific work and educational and organizational activities. The work included a list of 80 major scientific works of V. I. Bernik.
Bibliography: 31 titles.