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.
Bibliography: 20 titles.
