FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2002, VOLUME 8, NUMBER 2, PAGES 335-356

The variety N3N2 of commutative alternative nil-algebras of index 3 over a field of characteristic 3

A. V. Badeev

Abstract

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A variety is called a Specht variety if every algebra in this variety has a finite basis of identities. In 1981 S. V. Pchelintsev defined the topological rank of a Specht variety.

Let $\mathbf N_k$ be the variety of commutative alternative algebras over a field of characteristic $3$ with nilpotency class not greater than $k$. Let $\mathbf D$ be the variety $\mathbf N_3\mathbf N_2$ of nil-algebras of index $3$, i.e.\ the commutative alternative algebras with identities
$$
x^3=0,\quad [(x_1x_2)(x_3x_4)](x_5x_6)=0.
$$
 In the paper we prove that the varieties $\mathbf N_k\mathbf N_l$ are Specht varieties. Moreover, a base of the space of polylinear polynomials in the free algebra $F(\mathbf D)$ is built and the topological rank $\mathrm r_{\mathrm t}(\mathbf D_n)=n+2$ of varieties
$$
\mathbf D_n = \mathbf D \cap \mathrm{Var}((xy\cdot zt)x_1\ldots x_n)
$$
 is found. This implies that the topological rank of the variety $\mathbf D$ is infinite.

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Last modified: November 26, 2002