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: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=737866
• 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(保形场理论、弦理论等)越来越依赖于非常先进和抽象的数学，包括分析、几何和代数。我的研究旨在发现新的代数结构，这些结构从数学角度来看是有趣的，并具有应用于理论物理问题的潜力。我将研究这些代数结构的表示理论。表示是一个阴影，它仍然保存着关于这些结构的信息。在最基本的级别上，它是矩阵的集合，矩阵是数字的数组。实际上，这些结构本身通常可以使用矩阵来构建，但其条目不仅可以是数字，还可以是例如一个或几个变量的多项式。现在，这些代数结构有各种名称：李代数(纪念数学家索菲斯·李)、量子群、量子代数、延安、量子仿射代数等。在过去的三十五年里，延安和量子仿射代数一直是数学家和物理学家紧张研究的对象，一百多年来，延安和量子仿射代数一直是李代数的研究对象。表征理论的目标是通过关注这些结构的各种表征，对它们进行分类，将它们分解成更简单的块，确定基础和明确的、更具体的模型，从而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型扭曲延安的有限维、不可约表示的分类。这些都是更大规模表达的基石。分类应该根据具体的数据，如满足某些性质的多项式的元组。2.使用多线性代数、组合学等构建这些表示的显式模型。这可能允许我们找到基础并确定这些表示的维度。3.引入了新的B、C、D型扭曲量子仿射代数，其思想类似于我以前对扭曲延安的工作，并用某些多项式对它们的有限维不可约表示进行了分类。5.通过构造余积，研究表示范畴O，并通过函子将其与有限类型量子群的有限维表示范畴联系起来，发展了变形双流代数的表示理论。这里，范畴是集合概念的推广，函数器类似于函数，它将一个范畴中的一个表示赋给另一个范畴中的另一个表示，就像函数将一个数字赋给另一个数一样。6.利用量子场论的思想构造了与某些曲面相关的形变双流代数。7.建立了仿射延安与变形双流代数之间的联系。