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Sibirsk. Mat. Zh., 2013, Volume 54, Number 3, Pages 700–711 (Mi smj2452)  

This article is cited in 11 scientific papers (total in 11 papers)

Poisson algebras of polynomial growth

S. M. Ratseev

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

Abstract: Consider the sequence $c_n(V)$ of codimensions of a variety $V$ of Poisson algebras. We show that the growth of every variety $V$ of Poisson algebras over an arbitrary field is either bounded by a polynomial or at least exponential. Furthermore, if the growth of $V$ is polynomial then there is a polynomial $R(x)$ with rational coefficients such that $c_n(V)=R(n)$ for all sufficiently large $n$. We present lower and upper bounds for the polynomials $R(x)$ of an arbitrary fixed degree. We also show that the varieties of Poisson algebras of polynomial growth are finitely based in characteristic zero.

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

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English version:
Siberian Mathematical Journal, 2013, 54:3, 555–565

Bibliographic databases:

UDC: 512.572
Received: 24.05.2011

Citation: S. M. Ratseev, “Poisson algebras of polynomial growth”, Sibirsk. Mat. Zh., 54:3 (2013), 700–711; Siberian Math. J., 54:3 (2013), 555–565

Citation in format AMSBIB
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\by S.~M.~Ratseev
\paper Poisson algebras of polynomial growth
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\vol 54
\issue 3
\pages 700--711
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\vol 54
\issue 3
\pages 555--565
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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. O. I. Cherevatenko, “Mnogoobraziya lineinykh algebr polinomialnogo rosta”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 4(33) (2013), 7–14  mathnet  crossref  zmath  elib
    2. S. M. Ratseev, “O tozhdestvakh spetsialnogo vida v algebrakh Puassona”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 14:2 (2014), 150–155  mathnet
    3. 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
    4. S. M. Ratseev, “Correlation of Poisson Algebras and Lie Algebras in the Language of Identities”, Math. Notes, 96:4 (2014), 538–547  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    5. S. M. Ratseev, “Lie algebras with extremal properties”, Siberian Math. J., 56:2 (2015), 358–366  mathnet  crossref  mathscinet  isi  elib  elib
    6. M. V. Zaicev, D. Repovš, “Exponential growth of codimensions of identities of algebras with unity”, Sb. Math., 206:10 (2015), 1440–1462  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    7. S. M. Ratseev, “On minimal Poisson algebras”, Russian Math. (Iz. VUZ), 59:11 (2015), 54–61  mathnet  crossref
    8. S. M. Ratseev, O. I. Cherevatenko, “Korazmernosti mnogoobrazii algebr Puassona s lievo nilpotentnymi kommutantami”, Tr. IMM UrO RAN, 22, no. 1, 2016, 241–244  mathnet  mathscinet  elib
    9. S. M. Ratseev, “Proper $\mathrm{T}$-ideals of Poisson algebras with extreme properties”, Moscow University Mathematics Bulletin, 71:6 (2016), 224–232  mathnet  crossref  mathscinet  isi
    10. S. M. Ratseev, “Chislovye kharakteristiki mnogoobrazii algebr Puassona”, Fundament. i prikl. matem., 21:2 (2016), 217–242  mathnet
    11. A. Giambruno, S. Mishchenko, A. Valenti, M. Zaicev, “Polynomial codimension growth and the Specht problem”, J. Algebra, 469 (2017), 421–436  crossref  mathscinet  zmath  isi  scopus
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