A rational approach to affine quantum algebras

我们早期遇到的一个基本的数学运算是乘法；我们可以乘实数，有理数，整数，等等。这个操作有一个无穷无尽的美丽属性列表。例如，每一个正整数都可以唯一地写成质数的乘积，这一事实的应用超越了数学，例如在密码学中。然而，乘法也会在更抽象的环境中自然出现。在线性代数中，我们学会了矩阵的乘法——一组编码方程组的数字。进一步抽象，我们甚至可以将向量空间相乘，其中的例子包括实数线、平面、三维空间和高维类似物。我们为什么要进行这样的行动？事实证明，这种类型的空间乘法，被称为张量积，出现在许多显著的、具体的环境中；例如，在分析理论物理的某些晶格模型时，以及在研究重要的微分方程系统的量子类似物时。在这些设置中，底层的乘法是非常重要的，编码了一个丰富的对称列表，整个设置可以被视为产生于一个抽象的代数结构，通常被视为一种对称代数。正是在这种背景下，我研究的核心结构——量子群——出现了。更确切地说，我研究那些被称为仿射或双仿射型的量子代数的表示理论。在理论物理的量子逆散射方法的影响下，这些在20世纪80年代被正式引入，并从此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。这些结构的一个令人着迷的特点是，它们的许多关键属性可以用有理函数的语言优雅地描述。对于它们的张量结构来说尤其如此，这导致了仿射量子代数独特的大量有趣应用。我的研究目标是使用代数工具在几个新颖的方向上发展这种语言，然后应用它来解决开放的问题，并以有意义的方式扩展现有的理论。将讨论的具体问题的例子包括根据某些有理算子的奇异性研究张量积的整数素分解的一种变体，为仿射和扭曲杨量构造通用R和k矩阵（在量子可积性中出现的对象），以及发展新兴的仿射量子对称对理论。研究结果将对研究表示理论中各种主题的代数学者、对李论和颤振变体感兴趣的几何学者以及研究包括规范理论和量子可积系统在内的广泛主题的数学物理学家感兴趣。

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.