
This article is cited in 1 scientific paper (total in 1 paper)
Numerical characteristics of Leibniz–Poisson algebras
S. M. Ratseev^{a}, O. I. Cherevatenko^{b} ^{a} Ulyanovsk State University
^{b} Ulyanovsk State Pedagogical University
Abstract:
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 $\{x,y\}\cdot ż,t\}=0$, 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.
Keywords:
Poisson algebra, Leibniz algebra Leibniz–Poisson algebra, variety of algebras, growth of variety.
DOI:
https://doi.org/10.22405/222683832017181143159
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UDC:
512.572 Received: 12.11.2016 Accepted:13.03.2017
Citation:
S. M. Ratseev, O. I. Cherevatenko, “Numerical characteristics of Leibniz–Poisson algebras”, Chebyshevskii Sb., 18:1 (2017), 143–159
Citation in format AMSBIB
\Bibitem{RatChe17}
\by S.~M.~Ratseev, O.~I.~Cherevatenko
\paper Numerical characteristics of LeibnizPoisson algebras
\jour Chebyshevskii Sb.
\yr 2017
\vol 18
\issue 1
\pages 143159
\mathnet{http://mi.mathnet.ru/cheb539}
\crossref{https://doi.org/10.22405/222683832017181143159}
\elib{https://elibrary.ru/item.asp?id=29119842}
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This publication is cited in the following articles:

S. M. Ratseev, O. I. Cherevatenko, “O customaryprostranstvakh algebr Leibnitsa–Puassona”, Izv. Sarat. unta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 20:3 (2020), 290–296

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