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

Kac-Moody 量子对称对、KLR 代数和广义 Schur-Weyl 对偶性

基本信息

批准号：
EP/W022834/1
负责人：
Tomasz Przezdziecki
金额：
$ 38.19万
依托单位：
University of Edinburgh
依托单位国家：
英国
项目类别：
Fellowship
财政年份：
2023
资助国家：
英国
起止时间：
2023 至无数据
项目状态：
未结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FW022834%2F1
关键词：
Kac Moody quantum symmetric pairs

项目摘要

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.

对称性的概念是数学中最古老和最基本的概念之一。它起源于几何学；例如，正方形有八种对称性 - 四种反射和四种旋转。对称性有额外的结构：它们可以组合，并且在应用对称性之后，总是可以通过反对称性达到原始状态。这些性质在群的代数概念中被公理化。在我们的例子中，正方形的对称性产生了二面体群。我们所描述的过程也可以颠倒过来——给定一个群或另一个代数对象，我们可以将其更具体地实现为对称性的集合。这种实现称为表示。二十世纪初，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：对象集合的排列群（对称群）和可逆矩阵群（一般线性群）。尽管这些群体差异很大，但他们的代表本质上是相同的。这种关系现在被称为舒尔-韦尔对偶性，并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。该项目涉及这样一种概括，其起源于统计力学和量子场论。六顶点模型描述了二维冰样本中的氢键构型。该模型解背后的代数结构是著名的杨-巴克斯特方程，它本质上是编织群的表示。事实证明，这种表示与另一个称为量子群的对象的表示兼容。如果我们通过添加边界条件来丰富六顶点模型，那么杨-巴克斯特方程就被反射方程取代，量子群就必须升级为量子对称对，即由量子群及其共理想子代数组成的对。过去十年，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承认对量子对称对的推广。该项目的目标是使用各种代数和几何技术，研究 Schur-Weyl 对偶性背景下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——一种寻求用更通用的结构（如类别和函子）代替向量空间的方法。这就是为什么作为分类的基本工具的 Khovanov-Lauda-Rouquier 代数在该项目中发挥着核心作用。