The variety $\mathbf N_3\mathbf N_2$ of commutative alternative nil-algebras of index~3 over a~field of characteristic~$3$

A. V. Badeev

Fundamentalʹnaâ i prikladnaâ matematika, Tome 8 (2002) no. 2, pp. 335-356

Résumé

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.