Homological Techniques for Noncommutative Algebras and Tensor Categories

自然界中的许多物体，如花、水晶和分子，都表现出对称性。对称性可以通过运动来数学描述，例如，物体在旋转之后看起来是一样的。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，取而代之的是一个更适合量子现象的概念，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，这在当前量子计算和量子信息科学等数学应用中具有重要意义。该项目旨在开发和应用将量子对称性实例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有关量子对称性的问题的技术。该项目通过研究活动对学生进行培训。量子群、非对易代数和张量范畴是本项目中研究的量子对称的设置，它使用同调技术来阐明它们的结构。这项研究涉及几个方向。研究者将研究Hopf代数和张量范畴的上同调，特别是研究有限张量范畴的上同调总是有限生成的猜想。研究者的目的是建立关于有限张量范畴的支持簇结构的结果，利用同调技术与几何相结合来获得诸如张量积性质和野表示类型的判据等信息。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范畴中的循环，呼吁使用与A-无穷结构相关的技术。她还将继续开发改进的技术，以了解扭曲张量积代数，特别是斜群代数的Hochschild上同调的Lie结构。这一奖项反映了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。

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.