Far Eastern Mathematical Journal

To content of the issue

On minimal Leibniz – Poisson algebras of polynomial growth

S. M. Ratseev

2014, issue 2, P. 248–256

Abstract

Let $\{\gamma_n({\mathbf V})\}_{n\geq 1}$ be the sequence of proper codimension growth of a variety of Leibniz – Poisson algebras ${\mathbf V}$. We give one class of minimal varieties of Leibniz – Poisson algebras of polynomial growth of the sequence $\{\gamma_n({\mathbf V})\}_{n\geq 1}$, i.e. the sequence of proper codimensions of any such variety grows as a polynomial of some degree $k$, but the sequence of proper codimensions of any proper subvariety grows as a polynomial of degree strictly less than $k$.

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

Download the article (PDF-file)

References

[1] С. М. Рацеев, “Коммутативные алгебры Лейбница–Пуассона полиномиального роста”, Вестн. Сам. гос. ун-та. Естеств. серия, 94:3/1 (2012), 54–65. [2] Ю. А. Бахтурин, Тождества в алгебрах Ли, Наука, М., 1985. [3] A. Giambruno, M. V. Zaicev, Polynomial Identities and Asymptotic Methods, v. 122, Providence R.I., AMS Mathematical Surveys and Monographs, 2005. [4] Л. Е. Абанина, С. М. Рацеев, “Многообразие алгебр Лейбница, связанное со стандартными тождествами”, Вестн. Сам. гос. ун-та. Естеств. серия, 40:6 (2005), 36–50. [5] A. Giambruno, D. La Mattina, V. M. Petrogradsky, “Matrix algebras of polynomial codimention growth”, Israel J. Math., 158 (2007), 367–378. [6] D. La Mattina, “Varieties of almost polynomial growth: classifying their subvarieties”, Manuscripta Math., 123:2 (2007), 185–203. [7] S. P. Mishchenko, A. Valenti, “A Leibniz variety with almost polynomial growth”, J. Pure Appl. Algebra, 202:1-3 (2005), 82–101.

To content of the issue