Quantum Symmetries: tensor categories, braids, and Hopf algebras

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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.