Just-non-Lie nilpotent varieties of associative algebras

Details Just-non-Lie nilpotent varieties of associative algebras Just-non-Lie nilpotent varieties of associative algebras

2011-08-29

arxiv.org/abs/1108.5670v1

Mathematics

Rings and Algebras

12 pages

Scientific paper

We consider associative algebras over a field. An algebra variety is said to be {\em Lie nilpotent} if it satisfies a polynomial identity of the kind $[x_1, x_2, ..., x_n] = 0$ where $[x_1,x_2] = x_1x_2 - x_2x_1$ and $[x_1, x_2, ..., x_n]$ is defined inductively by $[x_1, x_2, ..., x_n]=[[x_1, x_2, ..., x_{n-1}],x_n]$. It easy to see that every non-Lie nilpotent variety contains a minimal such subvariety. In the case of characteristic zero a complete description of the minimal non-Lie nilpotent (i.e. {\em just-non-lie nilpotent}) varieties is found by Yu.Mal'cev. In the case of positive characteristic we reduce the problem of a description of such varieties to the case of {\em prime} varieties.

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