Hopf algebras, Frobenius-Schur indicators and modular categories

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The principal investigator proposes to study Hopf algebras, modular tensor categories, and some of their invariants called Frobenius-Schur indicators. In particular, the following topics will be studied. (1) Classification of Hopf algebras of small dimension: It is fundamentally important to understand some basic examples in any mathematical theory. The PI will continue to work on the classification of Hopf algebras whose dimension admits a simple factorization.(2) Gauge invariants of finite-dimensional Hopf algebras: It is generally difficult to decide whether two given finite-dimensional Hopf algebras have monoidally inequivalent representation categories. The PI expects to discover some invariants under Drinfeld twists, which are generalizations of Frobenius-Schur indicators to arbitrary finite-dimensional Hopf algebras. (3) Arithmetic properties of Frobenius-Schur indicators: The arithmetic properties of Frobenius-Schur indicators have led to the quantum Cauchy theorem for integral fusion categories and a congruence subgroup theorem for modular tensor categories. The PI will continue to explore these properties and their applications to fusion categories.The appeal of symmetry has been guidance for understanding the nature of science. Some symmetry can be described in terms of algebraic structures. For instance, the symmetry of platonic solids can be described by some finite groups of spatial rotations. The symmetry of some physical systems can be described by Hopf algebras, and tensor categories, which are closely related to other areas of mathematics such as representation theory, topological invariants, and conformal field theory. The proposed research studies some fundamental questions in the field, with an eye on applications in other areas.