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 Sib. Èlektron. Mat. Izv., 2015, Volume 12, Pages 500–507 (Mi semr605)

Mathematical logic, algebra and number theory

Complexity functions of some Leibniz–Poisson algebras

S. M. Ratseeva, O. I. Cherevatenkob

a Ulyanovsk State University, Lev Tolstoy, 42, 432017, Ulyanovsk, Russia
b Ulyanovsk State I.N.Ulyanov Pedagogical University, Ploshchad' 100-letiya so dnya rozhdeniya V.I. Lenina, 4, 432700, Ulyanovsk, Russia

Abstract: Leibniz–Poisson algebras are generalizations of Poisson algebras. Let $\{c_n(\mathbf{V})\}_{n\geq 0}$ and $\{\gamma_n(\mathbf{V})\}_{n\geq 2}$ are respectively sequences of codimensions and proper codimensions of varieties of Leibniz-Poisson algebras $\mathbf{V}$. We study the exponential generating functions $\mathcal{C}(\mathbf{V},z)=\sum_{n=0}^{\infty}c_n(\mathbf{V})z^n/n!$ and $\mathcal{C}^{p}(\mathbf{V},z)=\sum_{n=2}^{\infty}\gamma_n(\mathbf{V})z^n/n!$. The functions $\mathcal{C}(\mathbf{V},z)$ are used in the study of Lie algebras and associative algebras. In this paper we study numerical characteristics of varieties of Leibniz–Poisson algebras $\mathbf{V}_s$ defined by the identities
$$\{ x_1, x_2 \} \cdot \{x_3, x_4 \} =0, \{x_0,\{x_1,x_2\},\ldots ,\{x_{2s-1},x_{2s}\}\}=0$$
and of varieties of Leibniz–Poisson algebras $\mathbf{W}_s$ defined by the identities
$$\{ x_1, x_2 \} \cdot \{x_3, x_4 \} =0, \{\{x_1,x_2\},\ldots ,\{x_{2s+1},x_{2s+2}\}\}=0, s\geq 1.$$
For each of the variety $\mathbf{V}_s$ and $\mathbf{W}_s$ an algebra-carrier is found and a basis of $n$-th proper polylinear space is built. We found exact formulas for the exponential generating functions for the codimension sequences and for the proper codimension sequences and exact formulas for codimension and proper codimension. Also a series of varieties of Leibniz–Poisson algebras, which codimension sequences asymptotically grow as polynomials of degree $k$, $k \geq 2$, is given.

Keywords: Poisson algebra, Leibniz–Poisson algebra, variety of algebras, growth of variety.

DOI: https://doi.org/10.17377/semi.2015.12.042

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UDC: 512.572
MSC: 17B63
Received June 12, 2015, published September 10, 2015

Citation: S. M. Ratseev, O. I. Cherevatenko, “Complexity functions of some Leibniz–Poisson algebras”, Sib. Èlektron. Mat. Izv., 12 (2015), 500–507

Citation in format AMSBIB
\Bibitem{RatChe15} \by S.~M.~Ratseev, O.~I.~Cherevatenko \paper Complexity functions of some Leibniz--Poisson algebras \jour Sib. \Elektron. Mat. Izv. \yr 2015 \vol 12 \pages 500--507 \mathnet{http://mi.mathnet.ru/semr605} \crossref{https://doi.org/10.17377/semi.2015.12.042} `

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This publication is cited in the following articles:
1. S. M. Ratseev, O. I. Cherevatenko, “Chislovye kharakteristiki algebr Leibnitsa–Puassona”, Chebyshevskii sb., 18:1 (2017), 143–159
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