Affine and double affine quantum algebras

仿射和双仿射量子代数

• 批准号: RGPIN-2019-04799
• 负责人: Guay, Nicolas
• 金额: $ 1.53万
• 依托单位: University of Alberta
• 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=758342
• 关键词: Affine; double; affine; quantum; algebras

Theoretical physics (conformal field theory, string theory, etc.) relies increasingly on very advanced and abstract mathematics, including analysis, geometry and algebra. My research aims to uncover new algebraic structures that are interesting from a mathematical point of view and also hold the potential for applications to questions in theoretical physics. I will investigate the representation theory of those algebraic structures. A representation is a shadow which still holds information about those structures. At the most basic level, it is a collection of matrices, which are arrays of numbers. Actually, those structures themselves can often be built using matrices, but with entries that are not just numbers but could also be, for instance, polynomials in one or several variables.      Those algebraic structures go under various names: Lie algebras (in honour of the mathematician Sophus Lie), quantum groups, quantum algebras, Yangians, quantum affine algebras, etc. They have been the subject of intense research activity by both mathematicians and physicists, for the past thirty-five years in the case of Yangians and quantum affine algebras, for over a hundred years in the case of Lie algebras. The goal of representation theory is to understand those structures better by focusing on their various representations, classifying these, decomposing them into simpler blocks, determining bases and explicit, more concrete, models. The objectives of my research program include the following: 1.Complete the classification of finite dimensional, irreducible representations of twisted Yangians of types B-C-D. Those are building blocks for larger representations. The classification should be in terms of concrete data like tuples of polynomials satisfying certain properties. 2.Construct explicit models of those representations using multi-linear algebra, combinatorics, etc. This may allow us to find bases and determine the dimension of those representations. 3.Introduce new twisted quantum affine algebras of types B,C,D using ideas similar to those used in my past work for twisted Yangians and classify also their finite dimensional, irreducible representations in terms of certain polynomials. 5. Develop the representation theory of deformed double current algebras by constructing a coproduct, studying the category O of representations and connecting it via a functor to the category of finite dimensional representations of a quantum group of finite type. Here, a category is a generalization of the notion of set and a functor is similar to a function in that it assigns one representation in one category to another one in another category, the same way that a function assigns a number to another number. 6. Construct deformed double current algebras associated to certain surfaces using ideas from quantum field theory. 7. Establish connections between affine Yangians and deformed double current algebras.

理论物理学（共形场理论，弦理论等）越来越依赖非常先进和抽象的数学，包括分析，几何和代数。我的研究旨在揭示从数学角度有趣的新代数结构，这些结构也很有趣，并且还可以将其应用于理论物理学中的问题。我将研究这些代数结构的表示理论。表示形式是阴影，仍然包含有关这些结构的信息。在最基本的层面上，它是数字数组的收集。实际上，这些结构本身通常可以使用材料来构建，而且条目不仅是数字，而且可能是一个或几个变量中的多项式。这些代数结构以各种名称为单位：谎言代数（纪念数学家Sophus Lie），量子组，量子代数，扬吉人，量子代数等等。在过去的三十个案例中，在过去的三十个案例中，他们一直是数学家和物理学家的激烈研究活动的主题代数。表示理论的目的是通过专注于它们的各种表示，对这些结构进行分类，将它们分解为更简单的块，确定基础和明确，更具体的模型来更好地理解这些结构。我的研究计划的目标包括：1。完成B-C-D型扭曲的Yangians的有限维度，不可还原表示的分类。这些是用于更大表示形式的基础。该分类应根据具体数据，例如满足某些特性的多项式的元素。 2.使用多线性代数，组合主义者等构建这些表示的明确模型。这可能使我们3.使用与我过去用于扭曲的Yangians的工作相似的想法的新的扭曲量子仿射代数使用类似于我的最终尺寸，不可及其对某些多元型的不可及的表示的想法。 5。通过构建一个副反应，研究表示的类别O并通过函数将其连接到量子类型量子类型的有限维表示类别。在这里，类别是集合概念的概括，功能单元类似于一个函数，因为它在一个类别中为另一个类别中的一个类别分配了一个表示，就像函数将数字分配给另一个数字一样。 6。使用量子场理论中的思想与某些表面相关的构造变形双电流代数。 7。建立扬吉亚人与变形双电流代数之间的联系。