Abstract:
The problems related to the description of identities that hold in all $n$-dimensional associative nilpotent algebras over a field ($n$ is fixed) are studied. The author previously formulated the hypothesis that an arbitrary $n$-dimensional nilpotent algebra over any field satisfies some standard identity of minimal degree, and a number of results were obtained in support of this hypothesis. In this article, it turns out that this hypothesis is also confirmed in the class of $2$-algebras, that is, such locally nilpotent algebras over a field that the square of the principal ideal generated by any of the generators of the algebra is equal to zero. Moreover, the ideal of identities of a variety generated by $n$-dimensional $2$-algebras over an arbitrary field ($n$ is fixed) is described.
Keywords:variety, ideal of identities, nilpotent algebra.
Fulltext:
PDF file (324 kB)
First page:
PDF file