Quantum Symmetries: tensor categories, braids, and Hopf algebras

该奖项支持在一个可以用来在量子水平上模拟对称性的数学领域的研究。对称性存在于我们的日常生活中，它们是由称为群的对象进行数学编码的。近几十年来，量子现象引起了人们的极大关注。在这种背景下，量子对称性出现了，量子对称性最好地被称为张量范畴的类群对象所描述。此外，张量范畴是量子计算和物质拓扑相的语言。研究人员希望对模范畴(张量范畴的一种特殊类别)进行分类，因为它们在凝聚态物理和量子计算中的应用。特别地，酉模范畴是任意子的代数模型，它在拓扑量子计算中有应用。这个模型还具有作为任意子的容错模型的优势，因为微小的扰动不会改变量子系统的物理性质。张量范畴可以被认为是环的范畴版本。这个概念包括群、李代数和Hopf代数的表示。该项目将包括三个主要项目：张量范畴的分类、构造和同调性质的研究。其中一个目的是推进低阶预模范类和超模范畴的分类。这个项目的另一个目标是了解上同调在标准构造下的行为，并利用这些结果得到满足某些性质的Hopf代数和范畴的新例子，例如Etingof和Ostrik提出的关于Hopf代数和张量范畴的上同调的有限生成。调查者将探索支持度变化理论的概括。在有限张量范畴的背景下，这些是测量射影的几何对象。该项目的最后一个目标是使用Hopf代数和模张量范畴来寻找有限张量范畴的新结构，通过测量对称性并关注给定模范畴的Deligne乘积上的置换作用。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This award supports research in an area of mathematics that can be used to model symmetry at the quantum level. Symmetries are present in our daily life and they are encoded mathematically by objects called groups. In recent decades, a lot of attention has been given to quantum phenomena. In this context, quantum symmetries appear, which are best described by group-like objects called tensor categories. Moreover, tensor categories are the language of quantum computation and topological phases of matter. Researchers would like to classify modular categories (a distinguished class of tensor categories) because of their applications in condensed matter physics and quantum computing. In particular, unitary modular categories are algebraic models of anyons, which have applications in topological quantum computing. This model also has the advantage of being a fault-tolerance model of anyons, since small perturbations will not change the physical properties of the quantum system.Tensor categories can be thought of as the categorical version of rings. This notion includes representation of groups, Lie algebras, and Hopf algebras. The project will consist of three main programs: classification, construction, and study of homological properties of tensor categories. One of the aims is to advance on the classification of low-rank premodular categories and supermodular categories. Another goal of this project is to understand how the cohomology behaves under standard constructions and use these results to get new examples of Hopf algebras and categories satisfying certain properties, such as the finite generation of cohomology for Hopf algebras and tensor categories, proposed by Etingof and Ostrik. The investigator will explore generalizations of the support variety theory. These are geometrical objects that measure projectivity, in the context of finite tensor categories. The last goal of the project is to search for new constructions of finite tensor categories, using Hopf algebras, and modular tensor categories, via gauging the symmetry with a focus on permutation actions on Deligne products of a given modular category.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.