Indecomposable Lie algebra representations

不可分解的李代数表示

• 批准号: RGPIN-2020-04062
• 负责人: Szechtman, Fernando
• 金额: $ 1.31万
• 依托单位: University of Regina
• 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=759140
• 关键词: Indecomposable; Lie; algebra; representations

Lie algebras attract much attention due to their intrinsic beauty, importance, and deep connections to other areas of Mathematics and Mathematical Physics, such as differential geometry, particle physics, quantum groups, differential equations, simple groups of Lie type, etc. As is the case with most mathematical objects, Lie algebras are defined axiomatically. In addition to the initial objects that suggested the given axioms, many more suddenly spring to life, and one of the goals of the theory, albeit utopian, is to classify all possible Lie algebras by identifying as one all those that are exact mirror images of each other. Of great aid in this endeavor is Representation Theory, which studies all images (faithful or not) of a given Lie algebra; these consist of the various ways in which an abstract Lie algebra can act as a concrete Lie algebra of linear transformations of a vector space. There is a well behaved family of Lie algebras, called semisimple, whose (finite dimensional) representation theory is well understood in terms of atomic or irreducible representations, which can be combined to produce all others. However, the vast majority of Lie algebras are not semisimple and the atomic representations, now called indecomposable, are extremely difficult to understand. These are the ones I propose to investigate, building upon my prior work on the subject, while aiming deeper and further. Their understanding would constitute a significant contribution to our knowledge of the representation theory of Lie algebras in general, beyond the classical case of semisimple Lie algebras, and in area where not much is known at present. My research program will have a direct impact on the graduate and undergraduate university students I teach, including prospective teachers, as well as on the high school and elementary students with whom I interact through outreach activities. By being close to the frontier of knowledge, constantly pondering and solving research questions, and by working collaboratively with other active investigators, I am in a position to inspire and stimulate the future generation of critical thinkers and creators and help them reach their full potential. This can only be achieved by those instructors who are most active in their field of expertise. Canada will benefit by allocating resources so that its youth receive a top quality education.

李代数吸引了很多关注，由于其内在的美丽，重要性，以及与数学和数学物理学的其他领域，如微分几何，粒子物理，量子群，微分方程，李型简单群等的深刻联系，与大多数数学对象一样，李代数被公理化定义。除了最初提出这些公理的对象之外，还有更多的对象突然涌现出来，这个理论的目标之一，尽管是乌托邦式的，是通过将所有那些彼此精确镜像的李代数识别为一个来对所有可能的李代数进行分类。在这奋进有很大帮助的是表示论，它研究给定李代数的所有图像（忠实或不忠实）;这些包括抽象李代数可以作为向量空间的线性变换的具体李代数的各种方式。有一类行为良好的李代数，称为半单，其（有限维）表示理论在原子或不可约表示方面得到了很好的理解，它们可以组合起来产生所有其他的表示。然而，绝大多数李代数不是半单的，而现在被称为不可分解的原子表示非常难以理解。这些是我建议调查的，建立在我以前对这个问题的工作基础上，同时瞄准更深更远。他们的理解将构成一个显着的贡献，我们的知识表示理论的李代数一般，超越了经典的情况下，半单李代数，并在该地区没有多少是已知的目前。我的研究计划将对我教的研究生和本科生，包括未来的教师，以及我通过外展活动与之互动的高中和小学生产生直接影响。通过接近知识的前沿，不断思考和解决研究问题，并通过与其他积极的研究者合作，我能够激励和激发未来一代的批判性思想家和创造者，并帮助他们充分发挥潜力。这只能通过那些在他们的专业领域最活跃的导师来实现。加拿大将受益于分配资源，使其青年接受最高质量的教育。