Kac-Moody quantum symmetric pairs, KLR algebras and generalized Schur-Weyl duality

对称概念是数学中最古老、最基本的概念之一。它起源于几何学；例如，一个正方形有八种对称——四种反射和四种旋转。对称具有额外的结构：它们可以组合，并且在应用对称之后，总是可以通过逆对称达到原始状态。这些性质在群的代数概念中是公理化的。在我们的例子中，正方形的对称性产生了一个二面体群。我们所描述的过程也可以反过来——给定一个群或另一个代数对象，我们可以更具体地将其理解为对称的集合。这样的实现称为表示。20世纪初，Issai Schur和Hermann Weyl意识到两个非常重要的群的表示之间存在联系：对象集合的置换群（对称群）和可逆矩阵群（一般线性群）。尽管这两组人差别很大，但他们的表现在本质上是相同的。这种关系现在被称为Schur-Weyl二象性，并且构成了表征理论中最持久的主题之一，在许多不同的方向上有无数的概括。本项目关注的就是这样一个泛化，它起源于统计力学和量子场论。六顶点模型描述了二维冰样品中的氢键构型。这个模型的解背后的代数结构是著名的Yang-Baxter方程，它本质上是一个辫状群的表示。事实证明，这种表征与另一种被称为量子群的物体的表征是相容的。若在六顶点模型中加入边界条件，则将Yang-Baxter方程替换为反射方程，将量子群升级为量子对称对，即由量子群及其共理想子代数组成的对。在过去的十年里，人们对这一领域的兴趣激增，因为很明显，量子群论中熟悉的大多数结构都可以推广到量子对称对。本项目的目标是利用各种代数和几何技术，在Schur-Weyl对偶的背景下研究量子对称对的表示理论。我们方法的另一个重要组成部分是分类——一种寻求用更普遍的结构（如范畴和函子）取代向量空间的方法。这就是为什么分类的基本工具Khovanov-Lauda-Rouquier代数在该项目中发挥核心作用的原因。

The idea of symmetry is one of the oldest and most fundamental ones in mathematics. It has its origin in geometry; for example, a square has eight symmetries - four reflections and four rotations. Symmetries have extra structure: they can be composed, and after applying a symmetry one can always reach the original state via an inverse symmetry. These properties are axiomatized in the algebraic concept of a group. In our example, the symmetries of a square give rise to a dihedral group. The process we have described can also be reversed - given a group or another algebraic object, we can realize it more concretely as a collection of symmetries. Such a realization is called a representation. At the beginning of the twentieth century, Issai Schur and Hermann Weyl realized that there is a connection between the representations of two very important groups: the group of permutations of a collection of objects (the symmetric group) and the group of invertible matrices (the general linear group). Even though these groups are quite different, their representations are essentially the same. This relationship is now known as Schur-Weyl duality, and constitutes one of the most persistent themes in representation theory, with countless generalizations in many different directions. This project is concerned with one such generalization, whose origins are in statistical mechanics and quantum field theory. The six-vertex model describes the hydrogen-bond configurations in a two-dimensional sample of ice. The algebraic structure behind solutions to this model is the famous Yang-Baxter equation, which is, essentially, a representation of a braid group. It turns out that this representation is compatible with a representation of another object called a quantum group. If we enrich the six-vertex model by adding a boundary condition, the Yang-Baxter equation is replaced by the reflection equation, and the quantum group has to be upgraded to a quantum symmetric pair, i.e., a pair consisting of a quantum group and its coideal subalgebra. The last decade has seen an explosion of interest in this area, as it became clear that most structures familiar from quantum group theory admit a generalization to quantum symmetric pairs. The goal of this project is to study the representation theory of quantum symmetric pairs in the context of Schur-Weyl duality, using a variety of algebraic and geometric techniques. Another important component of our approach is categorification - a method which seeks to replace vector spaces by more universal structures like categories and functors. That is why Khovanov-Lauda-Rouquier algebras, a fundamental tool in categorification, play a central role in the project.