|
Sib. Èlektron. Mat. Izv., 2015, Volume 12, Pages 500–507
(Mi semr605)
|
|
|
|
This article is cited in 1 scientific paper (total in 1 paper)
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
Full text:
PDF file (160 kB)
References:
PDF file
HTML file
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}
Linking options:
http://mi.mathnet.ru/eng/semr605 http://mi.mathnet.ru/eng/semr/v12/p500
Citing articles on Google Scholar:
Russian citations,
English citations
Related articles on Google Scholar:
Russian articles,
English articles
This publication is cited in the following articles:
-
S. M. Ratseev, O. I. Cherevatenko, “Chislovye kharakteristiki algebr Leibnitsa–Puassona”, Chebyshevskii sb., 18:1 (2017), 143–159
|
Number of views: |
This page: | 131 | Full text: | 32 | References: | 34 |
|