Homological Techniques for Noncommutative Algebras and Tensor Categories

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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.