Lie algebras and superalgebras: representations and structure theory

李代数和超代数：表示和结构理论

• 批准号: RGPIN-2018-06417
• 负责人: Dimitrov, Ivan
• 金额: $ 2.33万
• 依托单位: Queen's University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758950
• 关键词: Lie; algebras; superalgebras; representations; structure

The idea of symmetry has played a central role in geometry, architecture, and art from ancient times. In the 1800's the notion of a group arose as the abstract algebraic structure encompassing all symmetries of a system. Towards the end of the 1800's, the Norwegian mathematician Sophus Lie disguised a class of groups which carries an additional structure reflecting the continuous structure of the system under consideration. These groups, called Lie groups, have become one of the most powerful tools in studying various geometric, analytic, and physical structures. Lie groups are highly non-linear but locally their structure is the same at every point. Thus, in many respects, studying a Lie group can be reduced to the simpler problem of studying a linear approximation, called a Lie algebra. The notion of a Lie superalgebra is a further generalization that traces its roots to the notion of supersymmetry in physics. Working with Z_2-graded objects, the mathematical terminology for “super-objects” in physics, is also the natural way to treat simultaneously commuting and anti-commuting qualities in mathematics. Thus, superalgebras provide the natural setting for problems arising in different field of mathematics, e.g. homological algebra. Lie algebras and Lie superalgebras - their structure and representation theory - have been studied extensively since the 1950's. Various classes of Lie (super)algebras and various classes of representations of Lie (super)algebras remain at the centre of research in Lie theory.The long-term goal of my research is two-fold: to study representations of infinite dimensional Lie algebras and to study the structure of Lie superalgebras and to obtain new results that unify and explain known phenomena for Lie algebras and Lie superalgebras. The current proposal is centred around four particular directions of research: 1. Weight representations of affine and finitary Lie algebras. 2. Left-symmetric superalgebras (LSSA). 3. Lagrangian subalgebras of simple Lie superalgebras. 4. Generalizations of root systems. I expect the outcomes of this research to have considerable impact in the areas of representation theory and combinatorics, as well as applications to physics. The proposal contains a number of projects suitable for training HQP at all levels. I intend to supervise and support three Ph.D. students, three M.Sc. students, five undergraduate students, and to co-supervise two postdoctoral fellows over the course of the grant.

自古以来，对称的概念在几何学、建筑学和艺术中就扮演着重要的角色。在1800年的概念出现了一个组作为抽象的代数结构包括所有对称性的系统。在19世纪末，挪威数学家Sophus Lie伪装了一类群，它们带有一个额外的结构，反映了所考虑系统的连续结构。这些群，称为李群，已经成为研究各种几何，分析和物理结构的最强大的工具之一。李群是高度非线性的，但局部上它们的结构在每一点上都是相同的。因此，在许多方面，研究李群可以简化为研究线性近似的更简单的问题，称为李代数。李超代数的概念是一个进一步的推广，其根源可以追溯到物理学中的超对称性概念。研究Z_2-阶化对象，物理学中的“超对象”的数学术语，也是数学中同时处理对易性和反对易性的自然方法。因此，超代数为不同数学领域中出现的问题提供了自然的背景，例如同调代数。李代数和李超代数-它们的结构和表示理论-自20世纪50年代以来得到了广泛的研究。各类李（超）代数和各类李（超）代数的表示仍然是李理论研究的中心。我研究的长期目标是双重的：研究无限维李代数的表示和研究李超代数的结构，并获得新的结果，统一和解释已知的李代数和李超代数的现象。目前的建议是围绕四个特定的研究方向：1。仿射和有限李代数的权表示。2.左对称超代数（LSSA）3.单李超代数的拉格朗日子代数。4.根系统的一般化。我希望这项研究的成果在表示论和组合学领域以及物理学应用方面产生相当大的影响。该建议书包含了一些适合培训各级HQP的项目。我打算指导和支持三个博士。学生，三个硕士生，五个本科生，并共同监督两个博士后研究员在赠款的过程中。