Groups, Rings, Lie and Hopf Algebras

代数是数学的基石之一。代数的对象：群、环、李代数、Hopf代数不仅应用于分析、几何、拓扑学等数学学科，而且应用于物理、化学、地质、生物等领域。在目前的形式中，这些对象是人类最敏锐的头脑长期发展和贡献的结果。它们崇高而抽象的形式使它们致力于解决现代科学中最困难的问题。他在代数方面的成就获得了最高奖项：菲尔兹奖和阿贝尔奖。这些奖项的获得者在我们感兴趣的领域做了重要的工作。菲尔兹奖得主泽尔马诺夫以解决涉及群、李和乔丹环的伯恩赛德问题而闻名。另一位菲尔兹奖得主，德林菲尔德，以他在数学物理学中涉及霍普夫代数的工作而闻名。阿贝尔奖获得者格罗莫夫最著名的定理之一是多项式增长群的特征。我在代数方面的工作始于45年前，当时我发表了一篇关于李代数的论文。从那时起，我已经发表了125多篇论文，涉及我目前提案的所有领域。其中一些包含了著名数学家提出的问题的解，比如马尔切夫关于有限李环中恒等式的有限基的问题。最近，我转向了关于标题对象的分类定理。没有人期望分类所有的组或环等。然而，这些物品往往带有一些自然属性。例如，了解具有这些性质的李代数的完整列表，可以极大地促进问题的解决，因为它使问题更加具体。例如，李代数上的群分级的分类对于数学物理中的收缩和超代数的研究，对于几何中出现的对称齐次空间的分类等等都是必不可少的。在对特征为非2的代数闭域上经典简单李代数上的阿贝尔群等级分类的竞争之后，我是主要贡献者之一，现在我们想要研究上面的应用，并将结果扩展到自然类：半简单的李代数，或局部有限简单的李代数，或模简单的李代数，或可解幂零的李代数。在这些课程中，已经迈出了令人鼓舞的第一步。因此，我们需要发展以前使用的代数群，群方案，Hopf代数和功能恒等式，以便将第一步转化为完全分类。除了分级，另一个研究领域对我们很重要：对群、李代数和超代数中各种类型的生长的研究。在无限群和环被证明在数学（分析，动力系统，几何和拓扑）和其他领域有用的情况下，这个增长是重要的。用生长函数代替代数的维数使人们能够定量地研究对象，否则只能定性地研究对象。许多名人过去和现在都在为团体的发展做出重要贡献，包括格里戈尔丘克、格罗莫夫、卡茨、奥尔尚斯基、泽尔马诺夫等人。与此同时，包括上述研究者在内的越来越多的研究人员正致力于将强大的群论方法推广到由生成器给出的代数和定义关系。这对于非交换几何和其他应用非常重要。例如，具有指数增长的同伦李代数出现在拓扑学和物理学中。最后，我将继续研究我的其他传统领域：pi代数，超代数和局部有限代数。

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.