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J. Sib. Fed. Univ. Math. Phys., 2013, Volume 6, Issue 1, Pages 97–104 (Mi jsfu293)  

This article is cited in 5 scientific papers (total in 5 papers)

On varieties of Leibniz–Poisson algebras with the identity $\{x,y\}\cdot ż,t\}=0$

Sergey M. Ratseev

Department of Mathematics and Information Technologies, Ulyanovsk State University, Ulyanovsk, Russia

Abstract: Let $K$ be an arbitrary field and let $A$ be a $K$-algebra. The polynomial identities satisfied by $A$ can be measured through the asymptotic behavior of the sequence of codimensions of $A$. 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 $\mathcal V$ does not contain any Leibniz polynomial identity then $\mathcal V$ has overexponential growth of the codimensions. We construct a variety of Leibniz–Poisson algebras with almost exponential growth.

Keywords: Poisson algebra, Leibniz–Poisson algebra, variety of algebras, growth of a variety.

Full text: PDF file (161 kB)
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UDC: 512.572
Received: 12.11.2012
Received in revised form: 12.11.2012
Accepted: 15.11.2012
Language:

Citation: Sergey M. Ratseev, “On varieties of Leibniz–Poisson algebras with the identity $\{x,y\}\cdot ż,t\}=0$”, J. Sib. Fed. Univ. Math. Phys., 6:1 (2013), 97–104

Citation in format AMSBIB
\Bibitem{Rat13}
\by Sergey~M.~Ratseev
\paper On varieties of Leibniz--Poisson algebras with the identity $\{x,y\}\cdot \{z,t\}=0$
\jour J. Sib. Fed. Univ. Math. Phys.
\yr 2013
\vol 6
\issue 1
\pages 97--104
\mathnet{http://mi.mathnet.ru/jsfu293}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. S. M. Ratseev, O. I. Cherevatenko, “O nekotorykh mnogoobraziyakh algebr Leibnitsa–Puassona s ekstremalnymi svoistvami”, Vestn. Tomsk. gos. un-ta. Matem. i mekh., 2013, no. 2(22), 57–59  mathnet
    2. S. M. Ratseev, O. I. Cherevatenko, “O tozhdestvakh spetsialnogo vida v algebrakh Leibnitsa–Puassona”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 2(35) (2014), 9–15  mathnet  crossref  zmath
    3. S. M. Ratseev, O. I. Cherevatenko, “Funktsii slozhnosti nekotorykh algebr Leibnitsa–Puassona”, Sib. elektron. matem. izv., 12 (2015), 500–507  mathnet  crossref
    4. A. R. Chekhlov, “On Abelian groups with commutative commutators of endomorphisms”, J. Math. Sci., 230:3 (2018), 502–506  mathnet  crossref  mathscinet  elib
    5. S. M. Ratseev, O. I. Cherevatenko, “Chislovye kharakteristiki algebr Leibnitsa–Puassona”, Chebyshevskii sb., 18:1 (2017), 143–159  mathnet  crossref  elib
  • Журнал Сибирского федерального университета. Серия "Математика и физика"
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