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: $ 1.17万
• 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: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=649308
• 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，对称的思想在几何、建筑和艺术中起着核心作用。在19世纪，群的概念作为包含系统所有对称性的抽象代数结构而出现。在19世纪末，挪威数学家Sophus Lie伪装了一类群，它带有一个额外的结构，反映了所考虑的系统的连续结构。这些群被称为李群，已经成为研究各种几何、解析和物理结构的最有力的工具之一。李群是高度非线性的，但其局部结构在每一点都是相同的。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概念。处理z_2级的物体，即物理学中“超级物体”的数学术语，也是同时处理数学中可交换性和反交换性的自然方法。因此，超代数为出现在不同数学领域的问题提供了自然的设置，例如同调代数。李代数和李超代数及其结构和表示理论自20世纪50年代以来得到了广泛的研究。李（超）代数的各种类型和李（超）代数的各种类型的表示一直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，研究李超代数的结构，并获得统一和解释李代数和李超代数已知现象的新结果。目前的提案主要围绕四个特定的研究方向：******1。仿射李代数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。根系的概括。******我希望这项研究的结果在表示理论和组合学领域产生相当大的影响，以及在物理学中的应用。******该建议包含一些适合培训各级HQP的项目。我打算在资助期间指导和支持3名博士生、3名硕士、5名本科生，并共同指导2名博士后。