Geodesic
On Leibniz–Poisson Special Polynomial Identities
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 135 (2014) no. 2, pp. 9-15 Cet article a éte moissonné depuis la source Math-Net.Ru

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In this paper we study Leibniz–Poisson algebras satisfying polynomial identities. We study Leibniz–Poisson special and Leibniz–Poisson extended special polynomials. We show that the sequence of codimensions $\{r_n({\mathbf V})\}_{n\geq 1}$ of every extended special space of variety ${\mathbf V}$ of Leibniz-Poisson algebras over an arbitrary field is either bounded by a polynomial or at least exponential. Furthermore, if this sequence is bounded by polynomial then there is a polynomial $R(x)$ with rational coefficients such that $r_n({\mathbf V}) = R(n)$ for all sufficiently large n. It follows that there exists no variety of Leibniz-Poisson algebras with intermediate growth of the sequence $\{r_n({\mathbf V})\}_{n\geq 1}$ between polynomial and exponential. We present lower and upper bounds for the polynomials $R(x)$ of an arbitrary fixed degree.
Keywords: Leibniz algebra, variety of algebras.
Mots-clés : Leibniz–Poisson algebra 
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S. M. Ratseev; O. I. Cherevatenko. On Leibniz–Poisson Special Polynomial Identities. Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 135 (2014) no. 2, pp. 9-15. http://geodesic.mathdoc.fr/item/VSGTU_2014_135_2_a18/

[1] S. M. Ratseev, “Commutative Leibniz–Poisson algebras of polynomial growth”, Vestnik SamGU. Estestvenno-Nauchnaya Ser., 2012, no. 3/1(94), 54–65 (In Russian)

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

[3] S. M. Ratseev, “Necessary and sufficient conditions of polynomial growth of varieties of Leibniz–Poisson algebras”, Russian Math. (Iz. VUZ), 58:3 (2014), 26–30 | DOI | Zbl

[4] S. M. Ratseev, O. I. Cherevatenko, “Exponents of some varieties of Leibniz—Poisson algebras”, Vestnik SamGU. Estestvenno-Nauchnaya Ser., 2013, no. 3(104), 42–52 (In Russian) | MR | Zbl

[5] S. M. Ratseev, “On exponents of some varieties of linear algebras”, Prikl. Diskr. Mat., 2013, no. 3, 32–34 (In Russian) | MR

[6] S. M. Ratseev, O. I. Cherevatenko, “On some varieties of Leibniz–Poisson algebras with extreme properties”, Vestn. Tomsk. Gos. Univ. Mat. Mekh., 2013, no. 2, 57–59 (In Russian)

[7] S. M. Ratseev, O. I. Cherevatenko, “On metabelian varieties of Leibniz–Poisson algebras”, IIGU Ser. Matematika, 6:1 (2013), 72–77 (In Russian) | Zbl

[8] O. I. Cherevatenko, “Varieties of linear algebras of polynomial growth”, Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki, 2013, no. 4(33), 7–14 (In Russian) | DOI | Zbl

[9] D. R. Farkas, “Poisson polynomial identities”, Comm. Algebra, 26:2 (1998), 401–416 | DOI | MR | Zbl

[10] D. R. Farkas, “Poisson polynomial identities II”, Arch. Math. (Basel), 72:4 (1999), 252–260 | DOI | MR | Zbl

[11] S. P. Mishchenko, V. M. Petrogradsky, A. Regev, “Poisson PI algebras”, Trans. Amer. Math. Soc., 359:10 (2007), 4669–4694 | DOI | MR | Zbl

[12] S. M. Ratseev, “Poisson algebras of polynomial growth”, Siberian Math. J., 54:3 (2013), 555–565 | DOI | MR | Zbl

[13] S. M. Ratseev, “Growth in Poisson algebras”, Algebra and Logic, 50:1 (2011), 46–61 | DOI | MR | Zbl | Zbl

[14] S. M. Ratseev, “Equivalent conditions of polynomial growth of a variety of Poisson algebras”, Moscow University Mathematics Bulletin, 67:5–6 (2012), 195–199 | DOI | MR | Zbl

[15] S. M. Ratseev, “On varieties of Poisson algebras with extremal properties”, Nauch. vedomosti BelGU. Mat. Fiz., 30:5(148) (2013), 107–110 (In Russian)

[16] S. M. Ratseev, “Estimates of the growth of certain varieties of Poisson algebras”, Nauch. vedomosti BelGU. Mat. Fiz., 31:11 (2013), 93–101 (In Russian)

[17] O. I. Cherevatenko, “On Lie nilpotent Poisson algebras”, Nauch. vedomosti BelGU. Mat. Fiz., 29:23(142) (2013), 14–16 (In Russian)

[18] I. P. Shestakov, U. U. Umirbaev, “The tame and the wild automorphisms of polynomial rings in three variables”, J. Amer. Math. Soc., 17:1 (2004), 197–227 | DOI | MR | Zbl

[19] M. Nagata, On the automorphism group on $k[x,y]$, Department of Mathematics, Kyoto University, Lectures in Mathematics, 5, Kinokuniya Book-Store Co., Tokyo, 1972, v+53 pp. | MR | Zbl

[20] Yu. A. Bakhturin, Tozhdestva v algebrakh Li [Identities in Lie Algebras], Nauka, Moscow, 1985, 448 pp. (In Russian) | MR | Zbl

[21] A. Giambruno, M. V. Zaicev, Polynomial Identities and Asymptotic Methods, Mathematical Surveys and Monographs, 122, American Mathematical Society, Providence R.I., 2005, xiv+352 pp. | DOI | MR | Zbl

[22] V. Drensky, Free algebras and PI-algebras. Graduate course in algebra, Springer-Verlag, Singapore, 2000, xii+271 pp. | MR | Zbl

[23] S. M. Ratseev, “Growth and colength of special type spaces of varieties of Poisson algebras”, Izvestiya vysshikh uchebnykh zavedeniy. Povolzhskiy region, 2006, no. 5(26), 125–135 (In Russian)

[24] V. Drensky, A. Regev, “Exact asymptotic behaviour of the codimention of some P.I. algebras”, Israel J. Math, 96:1 (1996), 231–242 | DOI | MR | Zbl

[25] S. M. Ratseev, O. I. Cherevatenko, “On the nilpotent Leibniz–Poisson algebras”, Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki, 2012, no. 4(29), 207–211 (In Russian) | DOI