Groups, Rings, Lie and Hopf Algebras

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Algebra is one of the cornerstones of Mathematics. The objects of Algebra: Groups, Rings, Lie algebras, Hopf algebras are used not only in the mathematical disciplines such as Analysis, Geometry, Topology, etc., but also in Physics, Chemistry, Geology, Biology and other areas. In the present form, these object are the result of very long development and contributions from the sharpest minds of the mankind. Their sublime and abstract form makes them working for the solution of the hardest problems in modern science. The achievements in Algebra have been marked by the highest awards: the Fields medals and the Abel prizes. The recipients of these awards did important work in the areas of interest to us. A Fields medalist Zelmanov is famous for his solution of the Burnside Problem involving Groups, Lie and Jordan rings. Another Fields medalist, Drinfeld, is famous for his work in Mathematical Physics involving Hopf algebras. One of the most celebrated theorems of an Abel Prize recipient Gromov characterizes groups of polynomial growth. My work in Algebra began 45 years ago, with a journal paper on varieties of Lie algebras. Since then I have published more than 125 papers in all areas in the title of my current proposal. Some of them contained solutions to problems raised by reputed mathematicians, such as Malcev’s problem on the finite basis of identities in finite Lie rings. Lately, I turned to the classification theorems concerning the entitled objects. No one expects classifying all groups or rings, etc. Yet often these objects come with some natural properties. Knowing the full list of, say, Lie algebras with these properties greatly facilitate the solution of problems, because it makes them much more specific. For example, the classification of gradings by groups on Lie algebras is essential for the study of contractions and superalgebras in Mathematical Physics, for the classification of the symmetric homogeneous spaces, appearing in Geometry, and so on. After the competition of the classification of abelian group gradings on classical simple Lie algebras over algebraically closed fields of characteristic not 2, where I was one of the main contributors, now we want to work on the applications, such as just above, and to the expand the results to natural classes: Lie algebras that are semisimple, or locally finite simple, or modular simple, or solvable and nilpotent. In each of these classes the first encouraging steps have been done. Hence we need to develop the used earlier: algebraic groups, group schemes, Hopf algebras and functional identities in order to transform the first steps into full classifications. Apart from gradings, another area of research will be important to us: the study of various types of growth in the groups, Lie algebras and superalgebras. The growth is important in the situation where infinite groups and rings prove useful in Mathematics (Analysis, Dynamical Systems, Geometry and Topology) and beyond. Replacing the dimension of an algebra by the growth function enables one to quantitatively study the objects, which otherwise could only be studied qualitatively. Many famous people did and keep doing important contributions to the growth of groups, including Grigorchuk, Gromov, Kac, Olshanskii, Zelmanov, and others. At the same time, more researchers including those above, are increasingly looking towards the extension of powerful group theoretic methods to algebras given by generators and defining relations. This is important for Noncommutative Geometry and other applications. For instance, the Homotopy Lie algebras with exponential growth appear in Topology and Physics. Finally, I will keep working in my other traditional areas: PI-algebras, superalgebras and locally finite algebras.