Groups, Rings, Lie and Hopf Algebras

群、环、李代数和 Hopf 代数

• 批准号: RGPIN-2014-04606
• 负责人: Bahturin, Yuri
• 金额: $ 1.02万
• 依托单位: Memorial University of Newfoundland
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=587909
• 关键词: Groups; Rings; Lie; Hopf; Algebras

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.

代数是数学的基石之一。代数的对象：群、环、李代数、霍普夫代数不仅用于分析、几何、拓扑等数学学科，还涉及物理、化学、地质、生物等领域。以目前的形式，这些物体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。 在代数的成就已被标记的最高奖项：菲尔兹奖章和阿贝尔奖。这些奖项的获得者在我们感兴趣的领域做了重要的工作。菲尔兹奖获得者泽尔曼诺夫是著名的解决方案的伯恩赛德问题涉及群体，李和约旦环。另一位菲尔兹奖获得者，德林费尔德，是著名的工作，在数学物理涉及霍普夫代数。 阿贝尔奖获得者格罗莫夫最著名的定理之一描述了多项式增长的群。 我在代数方面的工作始于45年前，当时我发表了一篇关于各种李代数的期刊论文。从那时起，我已经发表了超过125篇论文，在所有领域的标题，我目前的建议。 其中一些包含解决问题提出的著名数学家，如Malcev的问题的有限基础上的身份在有限李环。 最近，我转向了关于有标题的对象的分类定理。没有人期望对所有的组或环等进行分类，但这些对象通常具有一些自然属性。 比如说，知道具有这些性质的李代数的完整列表，会极大地促进问题的解决，因为它使问题更加具体。 例如，李代数上的群分次的分类对于数学物理中的压缩和超代数的研究，对于出现在几何中的对称齐性空间的分类，等等都是必不可少的。 在对特征不为2的代数闭域上的经典单李代数上的阿贝尔群分次分类的竞争之后，我是其中的主要贡献者之一，现在我们想研究应用，例如上面的应用，并将结果扩展到自然类：半单，或局部有限单，或模单，或可解和幂零的李代数。在这些课程中，第一个令人鼓舞的步骤已经完成。因此，我们需要发展以前使用的：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整的分类。 除了分级，另一个研究领域对我们来说也很重要：研究群、李代数和超代数中各种类型的增长。增长是重要的情况下，无限群和环证明有用的数学（分析，动力系统，几何和拓扑）和超越。 用增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。 许多著名的人做了，并继续做重要贡献的增长组，包括格里戈尔丘克，格罗莫夫，卡茨，Olshanskii，Zelmanov，和其他人。与此同时，越来越多的研究者，包括上面提到的，正越来越多地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。这对非交换几何和其他应用很重要。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，我将继续我的其他传统领域的工作：PI-代数，超代数和局部有限代数。