In this project Nikshych will investigate structure of tensor categories and their module categories from an algebraic point of view. Proposed problems include the theory of graded tensor categories and central extensions, study of module categories and categorical Morita equivalence, geometric analysis of braided categories, and classification of semisimple tensor categories and Hopf algebras. Most of the known examples of such categories appear from group-theoretical constructions (such as, e.g., equivariant sheaves on groups) and from representations of affine Lie algebras and quantum groups. Nikshych intends to extend and formalize these constructions, to develop categorical analogues of classical algebraic methods (extension theory of groups and Hopf algebras, Morita duality of rings, geometry of metric groups and metric Lie algebras), and to apply them to a structural study and classification of tensor categories.
Tensor categories are ubiquitous in representation theory and have applications in many areas of mathematics and physics. They are used to describe various non-commutative, or "quantum" symmetries, just like groups are used to describe classical symmetries. For instance, tensor categories appear in conformal field theory, in the study of invariants of knots and 3-manifolds, and in the theory of Jones-von Neumann subfactors in operator algebras. A recent conjecture of Boyarchenko and Drinfeld relates modular tensor categories with the theory of character sheaves. The main goal of Nikshych's research project is to study algebraic properties, structure, and classification of tensor categories and extend to a categorical setting the theory of Hopf algebras and quantum groups.
