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Algebra Logika, 2011, Volume 50, Number 1, Pages 68–88 (Mi al475)  

This article is cited in 7 scientific papers (total in 7 papers)

Growth in Poisson algebras

S. M. Ratseev

Ul'yanovsk State University, Ul'yanovsk, Russia

Abstract: A criterion for polynomial growth of varieties of Poisson algebras is stated in terms of Young diagrams for fields of characteristic zero. We construct a variety of Poisson algebras with almost polynomial growth. It is proved that for the case of a ground field of arbitrary characteristic other than two, there are no varieties of Poisson algebras whose growth would be intermediate between polynomial and exponential. Let $V$ be a variety of Poisson algebras over an arbitrary field whose ideal of identities contains identities
$$ \{\{x_1,y_1\},\{x_2,y_2\},…,\{x_m,y_m\}\}=0,\qquad\{x_1,y_1\}\cdot\{x_2,y_2\}\cdot\ldots\cdot\{x_m,y_m\}=0, $$
for some $m$. It is shown that the exponent of $V$ exists and is an integer.
For the case of a ground field of characteristic zero, we give growth estimates for multilinear spaces of a special form in varieties of Poisson algebras. Also equivalent conditions are specified for such spaces to have polynomial growth.

Keywords: Poisson algebra, growth of variety, colength of variety.

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English version:
Algebra and Logic, 2011, 50:1, 46–61

Bibliographic databases:

UDC: 512.572
Received: 22.11.2008
Revised: 20.04.2010

Citation: S. M. Ratseev, “Growth in Poisson algebras”, Algebra Logika, 50:1 (2011), 68–88; Algebra and Logic, 50:1 (2011), 46–61

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. S. M. Ratseev, O. I. Cherevatenko, “O nilpotentnykh algebrakh Leibnitsa–Puassona”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 4(29) (2012), 207–211  mathnet  crossref
    2. S. M. Ratseev, “Equivalent conditions of polynomial growth of a variety of Poisson algebras”, Moscow University Mathematics Bulletin, 67:5-6 (2012), 195–199  mathnet  crossref  mathscinet
    3. S. M. Ratseev, “Poisson algebras of polynomial growth”, Siberian Math. J., 54:3 (2013), 555–565  mathnet  crossref  mathscinet  isi
    4. S. M. Ratseev, O. I. Cherevatenko, “Eksponenty nekotorykh mnogoobrazii algebr Leibnitsa–Puassona”, Vestn. SamGU. Estestvennonauchn. ser., 2013, no. 3(104), 42–52  mathnet
    5. 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
    6. S. M. Ratseev, “Ob assotsiativnykh algebrakh slabogo rosta”, Vestn. SamGU. Estestvennonauchn. ser., 2014, no. 7(118), 70–74  mathnet
    7. S. M. Ratseev, “Chislovye kharakteristiki mnogoobrazii algebr Puassona”, Fundament. i prikl. matem., 21:2 (2016), 217–242  mathnet
  • Алгебра и логика Algebra and Logic
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