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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

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On Customary Spaces of Leibniz –Poisson Algebras

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

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.

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