On braided fusion categories I
Vladimir Drinfeld, Shlomo Gelaki, Dmitri Nikshych, Victor Ostrik
TL;DR
This work introduces the core of a braided fusion category as a new invariant that separates the finite-group (Tannakian) part from the remainder via de-equivariantization of a centralizer. It develops a self-contained framework for braided fusion categories without assuming pre-modularity or non-degeneracy, and it draws a quantum Casimir Lie algebra analogy to guide the theory. A key result is that the core, together with the induced group action, has an invariant class independent of the chosen maximal Tannakian subcategory, and non-degenerate cores correspond to centers of pointed categories. The paper also develops equivariantization/de-equivariantization, analyzes centralizers and projective centralizers, and introduces the lattice-theoretic core theory and weak anisotropy, laying a foundation for a structured approach to braided fusion categories and their centers in ongoing work.
Abstract
This work is a detailed version of arXiv:0704.0195 [math.QA]. We introduce a new notion of the core of a braided fusion category. It allows to separate the part of a braided fusion category that does not come from finite groups. We also give a comprehensive and self-contained exposition of the known results on braided fusion categories without assuming them pre-modular or non-degenerate. The guiding heuristic principle of our work is an analogy between braided fusion categories and Casimir Lie algebras.