Groups, Rings, Lie and Hopf Algebras

代数是数学的基石之一。代数的研究对象：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，而且还应用于物理、化学、地质、生物等领域。在目前的形式下，这些物品是人类最敏锐的头脑经过很长时间的发展和贡献的结果。它们的崇高和抽象的形式使它们致力于解决现代科学中最困难的问题。*代数的成就被最高奖项所标志：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。菲尔兹奖牌获得者泽尔马诺夫因解决涉及群、Lie和Jordan环的Burnside问题而闻名。另一位菲尔兹奖获得者，德因费尔德，因他在涉及霍普夫代数的数学物理方面的工作而闻名。阿贝尔奖获得者格罗莫夫最著名的定理之一刻画了多项式增长的群。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变种的期刊论文。从那时起，我在所有领域发表了125多篇论文，题目是我目前的提案。其中一些包含了著名数学家提出的问题的解决方案，例如有限李环中恒等式的有限基上的马尔科夫问题。*最近，我转向关于标题对象的分类定理。没有人期望对所有的群或环等进行分类。然而，这些对象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极大地促进了问题的解决，因为它使问题变得更加具体。*例如，李代数上的群的分次分类对于研究数学物理中的压缩和超代数、对称齐次空间的分类、出现在几何学中等等都是必不可少的。*在特征不为2的代数闭域上的经典单李代数上的交换群分次分类的竞争之后，我是主要贡献者之一，现在我们想要致力于应用，例如以上，并将结果扩展到自然类：半单的、局部有限的单的、模单的、可解的和幂零的李代数。在这些班级中，每一个班级都已经迈出了令人鼓舞的第一步。因此，我们需要发展前面使用的：代数群、群方案、Hopf代数和函数恒等式，以便将第一步转化为完全分类。*除了分级之外，另一个研究领域对我们来说将是重要的：研究群、李代数和超代数中的各种类型的增长。在无限群和环在数学(分析、动力系统、几何和拓扑学)和其他学科中被证明是有用的情况下，增长是重要的。用增长函数代替代数的维度使人们能够定量地研究对象，否则只能定性地研究对象。*许多名人曾经并将继续为团体的发展做出重要贡献，包括Grigorchuk、Gromov、Kac、Olshanskii、Zelmanov等。与此同时，越来越多的研究人员也在寻求将强大的群论方法扩展到由生成元和定义关系给出的代数中去。这对于非对易几何和其他应用程序很重要。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，我将继续在我的其他传统领域工作：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.