A rational approach to affine quantum algebras

仿射量子代数的理性方法

• 批准号: RGPIN-2022-03298
• 负责人: Wendlandt, Curtis
• 金额: $ 1.89万
• 依托单位: University of Saskatchewan
• 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=759166
• 关键词: rational; approach; affine; quantum; algebras

A basic mathematical operation we encounter early on is that of multiplication; we can multiply real numbers, rational numbers, integers, and so forth. This operation has an endless list of beautiful properties. For instance, every positive integer can be written uniquely as a product of prime numbers, a fact which has applications that transcend mathematics, such as in cryptography. Multiplication, however, also arises naturally in much more abstract settings. In linear algebra, one learns that we can multiply matrices - arrays of numbers which encode systems of equations. Abstracting this further, we can even multiply together vector spaces, examples of which include the real number line, the plane, three-dimensional space and higher dimension analogues. Why would we carry out an operation like this? It turns out that this type of multiplication of spaces, called a tensor product, arises in many remarkable, concrete settings; for instance, in analyzing certain lattice models of theoretical physics, and in the study of quantum analogues of important systems of differential equations. In these settings, the underlying multiplication is highly non-trivial, encoding a rich list of symmetries, and the entire setup may be viewed as arising from an abstract algebraic structure, often viewed as a type of symmetry algebra. It is in this context that the structures at the heart of my research, called quantum groups, arise. More precisely, I study the representation theory of those quantum algebras which are said to be of affine or double affine type. These were formally introduced in the 1980's, under the influence of the quantum inverse scattering method of theoretical physics, and have since become prolific mathematical objects whose theory frequently intertwines algebra, geometry and mathematical physics. A fascinating feature of these structures is that many of their key properties can be elegantly described in the language of rational functions. This is especially true for their tensor structure, and this leads to a wealth of interesting applications unique to affine quantum algebras. The goal of my research is to develop this language in several novel directions using algebraic tools, and then to apply it in order to address open problems and extend the existing theory in meaningful ways. Examples of specific problems that will be addressed include studying a variant of prime factorization of integers for tensor products in terms of the singularities of certain rational operators, constructing universal R and K-matrices (objects which arise in quantum integrability) for affine and twisted Yangians, and developing the emerging theory of affine quantum symmetric pairs. The results will be of interest to algebraists studying various topics in representation theory, geometers with interests in Lie theory and quiver varieties, and mathematical physicists studying a wide range of topics, including gauge theory and quantum integrable systems.

我们很早就遇到的基本数学操作是乘法。我们可以乘以实数，有理数，整数等。此操作有无尽的美丽属性列表。例如，每个积极的整数都可以独特地写成质数的产物，这一事实具有超越数学的应用，例如在密码学中。但是，在更抽象的设置中，乘法也自然出现。在线性代数中，人们了解到我们可以乘以矩阵 - 编码方程系统的数字数组。进一步抽象，我们甚至可以将矢量空间繁殖在一起，其中包括实际数字线，平面，三维空间和更高的维度类似物。我们为什么要进行这样的操作？事实证明，这种类型的空间乘法（称为张量产品）在许多引人注目的混凝土设置中出现。例如，在分析理论物理学的某些晶格模型以及在微分方程重要系统的量子类似过程中。在这些设置中，基础乘法是高度不平凡的，编码了丰富的对称性列表，并且整个设置可能被视为由抽象的代数结构引起的，通常被视为一种对称代数。正是在这种情况下，我的研究核心（称为量子群）出现了结构。更准确地说，我研究了那些据说具有仿射或双仿射类型的量子代数的表示理论。这些是在1980年代正式引入的，在理论物理学的量子反向散射方法的影响下，此后已成为多产的数学对象，其理论经常交织代数，几何和数学物理学。这些结构的一个引人入胜的特征是，可以用理性功能的语言优雅地描述它们的许多关键特性。它们的张量结构尤其如此，这导致了仿射量子代数独有的大量有趣应用。我的研究的目的是使用代数工具在几个新颖的方向上开发这种语言，然后应用它以解决开放问题并以有意义的方式扩展现有理论。将解决将解决的特定问题的示例包括研究张量产品的整数分解的变体，这些变体是根据某些理性运算符的奇异性来研究仿生和扭曲的Yangians的通用R和K-Matrices（在量子可集成性中产生的对象），并开发出差异量子对称对称的理论。在代表理论中研究各种主题的代数主义者，具有谎言理论和颤抖品种的兴趣的几何图形以及研究广泛主题的数学物理学家，包括量规理论和量子整合系统，将引起人们的兴趣。