Graphical and Categorical Methods in Representation Theory

这个项目是在一般领域的表示理论，这在广义上是理解一些自然发生的结构，如晶体的对称性的想法，通过研究这些对称性可以实现在矩阵方面的不同方式。通过同时寻找所有这些矩阵实现所共有的共同特征，数学家们发现了一些更高的结构，它们编码了对称之间所有可能的对称。这些新的结构，“monoidal categories”，往往可以实现在纯粹的图形术语。在过去的几年里，这种观点揭示了数学和物理学中非常不同的部分之间的一些非常意想不到的联系。在这个项目中发挥作用的一个例子是所谓的海森堡范畴，它最初出现在代数中，来自于对有限对称群表示之间的对称性的思考，但它也与无限维海森堡李代数有关，后者起源于量子力学。该项目的资助将通过利用这种图形方法，对表示论中的经典问题产生新的理解，并在数学内外都有潜在的应用，包括有限群论，组合学，李理论，纽结理论和理论物理。本计划为研究生提供研究训练的机会，更详细地说，本计划将通过利用一些由生成元和关系定义的新的关键monoidal范畴来研究代数群、量子群和李超代数的表示理论。这些包括各种海森堡范畴，其定义是图形性质的。这些monoidal范畴作用于表示论中的许多经典重要范畴，导致了一个统一的一般框架，揭示了不同理论之间意想不到的联系。该项目将集中在五个具体项目：Okounkov-Vershik方法表示的划分类别;拟遗传余代数和块的Deligne类别;合理的网络类别和加厚海森堡分类;奇海森堡和超级Kac-Moody 2-范畴;该奖项反映了美国国家科学基金会的法定使命，并已被认为是值得支持的，通过评估使用基金会的学术价值和更广泛的影响评审标准。

This project is in the general area of representation theory, which in broad terms is the idea of understanding symmetries of some naturally occurring structure such as a crystal by studying the different ways in which those symmetries can be realized in terms of matrices. By looking for the common features shared by all such matrix realizations at once, mathematicians have discovered some higher structures that encode all possible symmetries between symmetries. These new structures, "monoidal categories," can often be realized in purely graphical terms. In the last few years, this point of view has revealed some quite unexpected connections between very different parts of mathematics and physics. One example that plays a role in this project is the so-called Heisenberg category, which arose originally in algebra from thinking about symmetries between representations of the finite symmetric groups, but which is also connected to the infinite-dimensional Heisenberg Lie algebra, which has its origins in quantum mechanics. The funding of this project will lead to new understanding of classical problems in representation theory by taking advantage of this graphical approach, with potential applications both inside and outside of mathematics, including to finite group theory, combinatorics, Lie theory, knot theory, and theoretical physics. This project provides research training opportunities for graduate students.In more detail, the project will study the representation theory of algebraic groups, quantum groups and Lie superalgebras by exploiting some remarkable new pivotal monoidal categories which are defined by generators and relations. These include various Heisenberg categories whose definition is of a graphical nature. These monoidal categories act on many of the classically important categories in representation theory, leading to a unified general framework which reveals unexpected connections between different theories. The project will focus on five specific projects: Okounkov-Vershik approach to representations of the partition category; quasi-hereditary coalgebras and blocks of Deligne categories; the rational web category and thickened Heisenberg categorification; odd Heisenberg and super Kac-Moody 2-categories; Modular shifted Yangians and flag algebras.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.