ALMOST NILPOTENT VARIETIES IN DIFFERENT CLASSES OF LINEAR ALGEBRAS

A well founded way of researching the linear algebra is the study of it using the identities, consequences of which is the identity of nilpotent. We know the Nagata-Higman’s theorem that says that associative algebra with nil condition of limited index over a field of zero characteristic is nilpotent. It is well known the result of E.I.Zel’manov about nilpotent algebra with Engel identity. A set of linear algebras where a fixed set of identities takes place, following A.I. Maltsev, is called a variety. The variety is called almost nilpotent if it is not nilpotent, but each its own subvariety is nilpotent. Recently has been studied the growth of the variety. There is a variety of polynomial, exponential, overexponential growth, a variety with intermediate between polynomial and exponential growth. A variety has subexponential growth if it has polynomial or intermediate growth. This article is a review and description of almost nilpotent varieties in different classes of linear algebras over a field of zero characteristic. One part of the article is devoted to the case of classical linear algebras. Here we present the only associative almost nilpotent variety, it is the variety of all associative and commutative algebras. In the case of Lie algebras the almost nilpotent variety is the variety of all metabelian Lie algebras. In the case of Leibniz algebras we prove that there are only two examples of almost nilpotent varieties. All presented almost nilpotent varieties in this section have polynomial growth. In general case it was found that there are rather exotic examples of almost nilpotent varieties. In this work we describe properties of almost nilpotent variety of exponent 2, and also the existence of a discrete series of almost nilpotent varieties of different integer exponents is proved. The last section of the article is devoted to varieties with subexponential growth. Here we introduce almost nilpotent varieties for left-nilpotent varieties of index two, commutative metabelian and anticommutative metabelian varieties. As result we found that each of these classes of varieties contain exactly two almost nilpotent varieties.

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