This project will study the structure of certain finite-dimensional Hopf algebras, their categories of representations, and spherical fusion categories. The main part of this project concerns some categorical arithmetic invariants such as dimensions, exponents, Frobenius-Schur indicators, and representations of the modular group associated with modular tensor categories. These arithmetic invariants are closely related among each other, and they serve as important tools for studying these algebraic structures. A basic question is the relationship between the prime factors of the dimension of a simple object and those of the quasi-exponent of the underlying category. Any progress here would help in proving a conjecture of Kaplansky on semisimple Hopf algebras that remains open. The project also concerns the classification of Hopf algebras whose dimensions admit a simple prime factorization. Progress on the preceding basic question would also help in classifying finite-dimensional Hopf algebras.
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. Hopf algebras and tensor categories are generalizations of groups which can also describe the symmetry of some physical systems. They appear naturally in many areas of mathematics and mathematical physics such as conformal field theory, statistical mechanics, and quantum computations. Thus, a deeper understanding of these algebraic structures is of fundamental importance and it will eventually be useful in the other areas of science.
