Indecomposable Lie algebra representations

李代数因其内在的美、重要性以及与数学和数学物理的其他领域，如微分几何、粒子物理、量子群、微分方程、李型单群等的深刻联系而备受关注。与大多数数学对象一样，李代数是公理化定义的。除了提出给定公理的最初对象之外，还有更多的对象突然出现，而该理论的目标之一，尽管是乌托邦式的，是通过识别所有彼此完全镜像的李代数来分类所有可能的李代数。在这一努力中有很大帮助的是表征理论，它研究给定李代数的所有图像（忠实或不忠实）；这些包含了抽象李代数可以作为向量空间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，称为半简单代数，其（有限维）表示理论在原子或不可约表示方面被很好地理解，它们可以组合产生所有其他的。然而，绝大多数李代数都不是半简单的，而现在被称为不可分解的原子表示是非常难以理解的。这些都是我打算调查的，建立在我之前关于这个主题的工作的基础上，同时瞄准更深入和更远。他们的理解将构成一个重要的贡献，我们的知识的表示理论的李代数，超越经典的情况下，半单李代数，并在目前知之甚少的领域。我的研究项目将直接影响我所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生和小学生。通过接近知识的前沿，不断思考和解决研究问题，并与其他活跃的研究者合作，我能够激励和激发下一代批判性思想家和创造者，并帮助他们充分发挥潜力。这只能由那些在其专业领域最活跃的教师来实现。加拿大将通过分配资源使其青年接受高质量的教育而受益。

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.