On the scientific direction of the 'Lie algebras'

Galois theory, that with every equation of arbitrary degree in one unknown bound a finite group, and according to the structure of which it was possible to find if the equation could be solved in radicals or not, was already quite common by the end of the 19th century. Norwegian mathematician Sophus Lie decided to construct a similar theory for partial differential equations. He succeeded in constructing such theory, i.e. to compare to some differential equations a group by which it was possible to determine the solvability of the initial equation in radicals. Although this theory did not give an answer on the solvability for every differential equation. But at the same time there appeared groups that were very interesting. These groups are now called Lie groups. Lie groups are a special case of continuous groups (the latter by definition have the properties of two structures: the structure of a group and the structure of a topological space, and these structures are linked by the condition of continuity of group operations). The condition of continuity in the Lie groups was replaced by analyticity.

Each Lie group defines a Lie algebra (originally called the infinitesimal group), which determines the corresponding group locally (i.e., two Lie groups with coinciding Lie algebras are isomorphic locally, in other words they have the same neighborhood of the one). Thus the Lie algebras arose. Initially, the theory of Lie algebras have been studied in the (with) the theory of Lie groups. However, by the middle of the last (20th) century it has been issued as a separate field.

The main results in the theory of Lie algebras have been obtained by such famous mathematicians (apart from C. Lee) as: W. Killing, E. Cartan, H. Weyl, van der Waerden, A. Borel, Chevalley.

In Kazan the theory of Lie algebras began being studied in the 30s of last century. First of all the works of the famous algebraist N. Chebotarev and his students I.D. Ado, V. Morozov should be noted here. I.D. Ado has proved the existence of a precise finite-dimensional representation for any finite-dimensional Lie algebra over a field of characteristic zero and V. Morozov has proved an important theorem of regularity, which states that any not semisimple maximal subalgebra of a simple Lie algebra is regular (i.e., contains the so-called Cartan subalgebra). In addition, somewhat less significant results have been proved by him, for example, a theorem on the nilpotent element and the theorem that bears the name of the Borel-Morozov.

Note that the Lie algebra is defined as a linear space on which a cross product is set (an example of Lie algebra is the usual three-dimensional space over the real field to the vector product). Thus, the Lie algebra depends on the field (as opposed to group), over which it is defined.

By the middle of last century many fundamental questions in the theory of Lie algebras over fields of characteristic zero have been resolved and Lie algebras over fields of positive characteristic have become the objects of study. The so-called modular Lie algebra. In the late 70's A.I. Kostrikin and I.R. Shafarevich described all the available examples of such algebras, using a single structure of algebras of Cartan type, and hypothesized about the classification of modular Lie algebras.

In the resolving of this task members of our department took part. A.H. Dolotkazin obtained a des­cription of the simple modular Lie algebras of rank one. J.B. Ermolaev classified monogenic Lie algebra (appendix) using the technique of Cartan extensions with graduation of sufficient length. N.A. Koreshkov described the irreducible representations of restricted simple Lie algebras of Cartan type. S.M. Scryabin classified differential forms defining the algebras of Cartan type.

Currently, the classification of simple modular Lie algebras over fields of characteristic p is greater than or equal to 7 is completed and the studies are carried out in some classes that generalize the class of Lie algebras or are related with it by some constructions.