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.

对称性的想法在古代的几何，建筑和艺术中发挥了核心作用。在1800年代，一个群体的概念出现了，作为包含系统所有对称性的抽象代数结构。在1800年代结束时，挪威数学家索菲斯（Sophus）掩盖了一类群体，该组带有一个额外的结构，反映了所考虑的系统的连续结构。这些称为谎言组的群体已成为研究各种几何，分析和物理结构的最强大工具之一。谎言组是高度非线性的，但在本地的结构在每个点都是相同的。在许多方面，研究谎言组可以简化为研究线性近似（称为谎言代数）的简单问题。谎言超级甲虫的概念是进一步的概括，它的根源是物理学中超对称性的概念。使用Z_2毕业的对象，物理学中的“超级对象”的数学术语也是同时处理数学中同时通勤和反交换品质的自然方法。因此，超甲虫为在不同数学领域（例如同源代数。自1950年代以来，已经对谎言代数和谎言超级堡（其结构和代表理论）进行了广泛的研究。各个类别的谎言（超级）代数和谎言（超级）代数的各种类别的代数仍然存在于谎言理论的中心。我的研究的长期目标是两个方面：研究无限维度谎言代数的表示形式，并研究谎言超级日期的结构，并获得统一和解释已知现象的新结果，以撒谎和撒谎，撒谎和撒谎。当前的提议集中在研究的四个特定方向上：1。仿射和限制代数的权重表示。 2。左对称超级级（LSSA）。 3。简单的Like Superalgebras的Lagrangian子代理。 4。根系的概括。我希望这项研究的结果对代表理论和组合学领域以及对物理的应用有很大影响。该提案包含许多适合各个级别培训HQP的项目。我打算监督和支持三个博士学位。学生，三个硕士学生，五名本科生，并在赠款过程中共同享受两个博士后研究员。