Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)

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Ratseev S. M., Cherevatenko O. I. On Customary Spaces of Leibniz –Poisson Algebras. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2020, vol. 20, iss. 3, pp. 290-296. DOI: 10.18500/1816-9791-2020-20-3-290-296, EDN: HSODCH

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
31.08.2020
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Language: 
Russian
Heading: 
Mathematics
Article type: 
Article
UDC: 
512.572
DOI: 
10.18500/1816-9791-2020-20-3-290-296
EDN: 
HSODCH

On Customary Spaces of Leibniz –Poisson Algebras

Autors: 
Ratseev Sergey Mihailovich, Ulyanovsk State University
Cherevatenko Olga I., Ulyanovsk State I. N. Ulyanov Pedagogical University
Abstract: 

Let K be a base field of characteristic zero. It is well known that in this case all information about varieties of linear algebras V contains in its polylinear components Pn(V), n ∈ N, where Pn(V) is a linear span of polylinear words of n different letters in a free algebra K(X,V). D. Farkas defined customary polynomials and proved that every Poisson PI-algebra satisfies some customary identity. Poisson algebras are special case of Leibniz –Poisson algebras. In the paper the sequence of customary spaces of the free Leibniz –Poisson algebra {Q2n}n⩾1 is investigated. The basis and dimension of spaces Q2n are given. It is also proved that in case of a base field of characteristic zero any nontrivial identity of the free Leibniz –Poisson algebra has nontrivial identities in customary spaces.

Key words: 
Poisson algebra
Leibnitz –Poisson algebra
variety of algebras
growth of variety
References: 
  1. Ratseev S. M. Numerical characteristics of varieties of Poisson algebras. J. Math. Sci., 2019, vol. 237, iss. 2, pp. 304–322.
  2. Ratseev S. M., Cherevatenko O. I. Numerical characteristics of Leibniz – Poisson algebras. Chebyshevskii Sbornik, 2017, vol. 18, no. 1, pp. 143–159 (in Russian).
  3. Bahturin Yu. A. Identical relations in Lie algebras. Utrecht, VNU Science Press, 1987. 310 p. (Russ. ed.: Moscow, Nauka, 1985. 448 p.).
  4. Giambruno A., Zaicev M. V. Polynomial Identities and Asymptotic Methods. AMS Mathematical Surveys and Monographs. Vol. 122. Providence R.I., 2005. 352 p.
  5. Drensky V. Free algebras and PI-algebras: Graduate course in algebra. Singapore, Springer-Verlag, 2000. 271 p.
  6. Mishchenko S. P., Petrogradsky V. M., Regev A. Poisson PI algebras. Trans. Amer. Math. Soc., 2007, vol. 359, no. 10, pp. 4669–4695. DOI: https://doi.org/10.1090/S0002-9947-07-04008-1
  7. Farkas D. R. Poisson polynomial identities. Comm. Algebra, 1998, vol. 26, no. 2, pp. 401–416.
  8. Farkas D. R. Poisson polynomial identities II. Arch. Math., 1999, vol. 72, iss. 4, pp. 252–260. DOI: https://doi.org/10.1007/s000130050329
  9. Kaygorodov I. Algebras of Jordan brackets and generalized Poisson algebras. Linear and Multilinear Algebra, 2017, vol. 65, iss. 6, pp. 1142–1157. DOI: https://doi.org/10.1080/03081087.2016.1229257
  10. Kolesnikov P., Makar-Limanov L., Shestakov I. The Freiheitssatz for generic Poisson algebras. SIGMA, 2014, vol. 10, iss. 115, 15 p. DOI: https://doi.org/10.3842/SIGMA.2014.115
Received: 
20.05.2019
Accepted: 
09.09.2019
Published: 
31.08.2020
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