Geodesic
On minimal Poisson algebras
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 11 (2015), pp. 64-72.

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We give Poisson algebra constructions on the base of associative algebras and we explore the relationship between the obtained Poisson algebras and original associative algebras. Based on the obtained structures we give two classes of minimal varieties of Poisson algebras of polynomial growth, i.e., the sequence of codimensions of any such a variety grows as a polynomial of some degree $k$, but the sequence of codimensions of any proper subvariety grows as a polynomial of degree strictly less than $k$. Previously the author showed that there are only two varieties of Poisson algebras of almost polynomial growth. In this paper we give a complete description of all subvarieties of these two varieties.
Mots-clés : Poisson algebra
Keywords: associative algebra, variety of algebras, growth of a variety. 
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S. M. Ratseev. On minimal Poisson algebras. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 11 (2015), pp. 64-72. http://geodesic.mathdoc.fr/item/IVM_2015_11_a5/

[1] Bakhturin Yu. A., Tozhdestva v algebrakh Li, Nauka, M., 1985 | MR

[2] Ratseev S. M., “Algebry Puassona polinomialnogo rosta”, Sib. matem. zhurn., 54:3 (2013), 700–711 | MR | Zbl

[3] Regev A., “Existence of identities in $A\otimes B$”, Israel J. Math., 11 (1972), 131–152 | DOI | MR | Zbl

[4] Bahturin Yu., Mishchenko S., Regev A., “On the Lie and associative codimensions growth”, Commun. Algebra, 27:10 (1999), 4901–4908 | DOI | MR | Zbl

[5] Berele A., Regev A., “Hook Young diagrams with applications to combinatorics and to representations of Lie superalgebras”, Adv. Math., 64:2 (1987), 118–175 | DOI | MR | Zbl

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

[7] Ratseev S. M., “Ekvivalentnye usloviya polinomialnosti rosta mnogoobrazii algebr Puassona”, Vestn. Mosk. un-ta. Ser. 1. Matem. Mekhan., 67:5 (2012), 8–13 | MR | Zbl

[8] Drensky V., “Relations for the cocharacter sequences of $T$-ideals”, Proc. Intern. Conf. on Algebra Honoring A. Malcev, Part 2, Contemp. Math., 131, Part 2, 1992, 285–300 | DOI | MR | Zbl

[9] La Mattina D., “Varieties of almost polynomial growth: classifying their subvarieties”, Manuscripta Math., 123:2 (2007), 185–203 | DOI | MR | Zbl

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

[11] Ratseev S. M., “Neobkhodimye i dostatochnye usloviya polinomialnosti rosta mnogoobrazii algebr Leibnitsa–Puassona”, Izv. vuzov. Matem., 2014, no. 3, 33–39 | Zbl