On new properties of some varieties with almost polynomial growth

Yu. R. Pestova

Čebyševskij sbornik, Tome 16 (2015) no. 2, pp. 186-207

Résumé

In the study of different mathematical structures well known and long used in mathematics algebraic method is the selection of classes of objects by means of identities. Class of all linear algebras over some field in which a fixed set of identities takes place is called the variety of linear algebras over a given field by A.I. Malcev. We have such concept as the growth of the variety. There is polynomial or exponential growth in mathematical analysis. In this work we will speak about properties of some varieties in different classes of linear algebras over zero characteristic field with almost polynomial growth. That means that the growth of the variety is not polynomial, but the growth of any its own subvariety is polynomial. The article has a synoptic and abstract character. One unit of the article is devoted to the description of basic properties all associative, Lie's and Leibniz's varieties over zero characteristic field with almost polynomial growth. In the case of associative algebras there are only two such varieties. In the class of Lie algebras there are exactly four solvable varieties with almost polynomial growth and is found one unsolvable variety wiht almost polynomial growth and the question about its uniqueness is opened in our days. In the case of Leibniz algebras there are nine varieties with almost polynomial growth. Five of them are named before Lie varieties, which are Leibniz varieties too. The last four ones are varieties which have the same properties as solvable Lie varieties of almost polynomial growth. Next units we'll devote to famous and new characteristics of two Lie's varieties with almost polynomial growth. In the first of them we speak about found by us colength of the variety generated by three-dimensional simple Lie algebra $sl_2$, which is formed by a set of all $2\times2$ matrices with zero trace over a basic field with operation of commutation. Then it will be described a basis of multilinear part of the variety which consists of Lie algebras with nilpotent commutant degree not higher than two. Also we'll give formulas for its colength and codimension. The last unit is devoted to description the basis of multilinear part of Leibniz variety with almost polynomial growth defined by the identity $x_1(x_2x_3)(x_4x_5)\equiv0$.

Détail

Keywords: variety, almost polynomial growth, colength, codimension, basis of multilinear part.