Braided Fusion Categories - Lecture 1

Explore the algebraic theory of semisimple braided tensor categories, known as fusion categories, in this comprehensive 57-minute lecture by Dmitri Nikshych at the Centre de recherches mathématiques (CRM). Delve into the structure and classification of these categories, which naturally emerge in quantum group representation theory and provide an algebraic framework for describing various non-commutative or "quantum" symmetries. Examine applications in topological quantum field theories, Galois theory for Jones-von Neumann subfactors, and topological phases of matter. Cover key topics including basic properties and examples, centralizers and non-degeneracy, (de)-equivariantizations and the core of braided fusion categories, gauging and Picard groups, and Witt equivalence of non-degenerate categories. Designed to be accessible to graduate students, this lecture is part of the Workshop on Quantum symmetries: Tensor Categories, Topological quantum field theories, and Vertex algebras held in October-November 2022.