Affine and double affine quantum algebras

理论物理学（共形场论、弦理论等）越来越依赖于非常高级和抽象的数学，包括分析，几何和代数。我的研究旨在发现新的代数结构，从数学的角度来看是有趣的，也有可能应用到理论物理学的问题。我将研究这些代数结构的表示理论。一个表示是一个阴影，它仍然保存着关于这些结构的信息。在最基本的层次上，它是一个矩阵的集合，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，但条目不仅是数字，还可以是一个或多个变量的多项式。 这些代数结构下去各种名称：李代数（在荣誉的数学家索菲斯李），量子群，量子代数，Yangians，量子仿射代数等，他们一直是主题的激烈研究活动的数学家和物理学家，在过去35年的情况下，Yangians和量子仿射代数，超过一百年的情况下李代数。表征理论的目标是更好地理解这些结构，方法是关注它们的各种表征，对它们进行分类，将它们分解为更简单的块，确定基础和明确的，更具体的模型。 本论文的研究目标包括：1.完成B-C-D型扭杨格数的有限维不可约表示的分类。这些都是构建更大的表现。分类应根据具体的数据，如满足某些性质的多项式元组。2.使用多线性代数，组合学等构建这些表示的显式模型，这可以让我们找到基础并确定这些表示的维度。 3.引入新的扭曲量子仿射代数类型B，C，D使用类似的想法在我过去的工作中使用的扭曲杨格尔斯和分类也有限维，不可约表示在某些多项式。5.发展变形双流代数的表示理论，通过构造一个余积，研究表示的范畴O，并通过函子将其连接到有限类型量子群的有限维表示的范畴。在这里，范畴是集合概念的推广，函子与函数类似，因为它将一个范畴中的一个表示分配给另一个范畴中的另一个表示，就像函数将一个数分配给另一个数一样。6.用量子场论的思想构造与某些曲面相关的变形双流代数。7.建立仿射杨氏代数和变形双流代数之间的联系。

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.