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On customary spaces of Leibniz-Poisson algebras. (Russian. English summary) Zbl 1477.17033

Summary: Let \(K\) be a base field of characteristic zero. It is well known that in this case all information about varieties of linear algebras \(\mathbf{V}\) contains in its polylinear components \(P_n(\mathbf{V})\), \(n \in \mathbb{N} \), where \(P_n(\mathbf{V})\) is a linear span of polylinear words of \(n\) different letters in a free algebra \(K(X,\mathbf{V})\). D. R. Farkas [Commun. Algebra 26, No. 2, 401–416 (1998; Zbl 0892.17001)] 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 \(\{Q_{2n}\}_{n\geq 1}\) is investigated. The basis and dimension of spaces \(Q_{2n}\) 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.

MSC:

17A32 Leibniz algebras
17B63 Poisson algebras

Citations:

Zbl 0892.17001
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References:

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