Homological Techniques for Noncommutative Algebras and Tensor Categories

自然界中的许多物体，如花、晶体和分子，都具有对称性。对称性可以通过运动进行数学描述，例如，物体旋转后看起来是一样的。这样的运动共同形成了所谓的对称群。在量子物理学中，这个经典的对称群概念不再足以捕捉所有的对称性，而被一个更适合量子现象的概念所取代，称为量子对称群，或相关的更一般的数学结构。该项目开展量子对称性的基础研究，这在量子计算和量子信息科学等当前数学应用中非常重要。该项目旨在开发和应用将量子对称性分解为更小组件的技术，以促进理解。这项工作将扩展可用于回答应用中出现的量子对称性问题的技术。该项目涉及通过参与研究对学生进行培训。量子群，非交换代数和张量范畴是本项目研究的量子对称性的背景，它使用同调技术来阐明它们的结构。研究涉及几个方向。研究者将研究Hopf代数和张量范畴的上同调，特别是研究一个猜想，即有限张量范畴的上同调总是可生成的。研究人员的目标是建立有限张量类别的支持品种的结构上的结果，利用同调技术结合几何获得信息，如张量积属性和野生表示类型的标准。她计划在最近开发的两种不同技术之间建立直接联系，即同伦提升方法和精确类别中的循环，并调用与A-无穷结构有关的技术。她还将继续开发改进的技术，以理解扭曲张量积代数的Hochschild上同调的Lie结构，特别是斜群代数。该奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

Many objects in nature, such as flowers, crystals, and molecules, exhibit symmetry. Symmetry can be described mathematically through motions, for example, a rotation after which the object appears the same. Such motions collectively form what is called a symmetry group. In quantum physics, this classical notion of symmetry group is no longer enough to capture all symmetries and is replaced by a notion more suited to quantum phenomena, termed quantum symmetry groups, or by related more general mathematical structures. This project carries out basic research in quantum symmetry, which is important in current applications of mathematics such as quantum computing and quantum information science. The project aims to develop and apply techniques for breaking down instances of quantum symmetry into smaller components, facilitating understanding. This work will expand the techniques available for answering questions about quantum symmetry that arise in applications. The project involves training of students through research involvement.Quantum groups, noncommutative algebras, and tensor categories are the settings for quantum symmetry studied in this project, which uses homological techniques to shed light on their structure. The research involves several directions. The investigator will study the cohomology of Hopf algebras and tensor categories, specifically to investigate a conjecture that cohomology of finite tensor categories is always finitely generated. The investigator aims to establish results on the structure of support varieties for finite tensor categories, harnessing homological techniques combined with geometry to obtain information such as a tensor product property and criteria for wild representation type. She plans to establish a direct connection between two different techniques that were developed recently, the homotopy lifting method and loops in exact categories, calling on techniques used in connection with A-infinity structures. She will also continue to develop improved techniques for understanding the Lie structure of the Hochschild cohomology of twisted tensor product algebras, particularly skew group 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.