In arXiv:1910.12059 Liu, Palcoux and Wu proved a remarkable necessary condition for a fusion ring to admit a unitary categorification, by constructing invariants of the fusion ring that have to be positive if it is unitarily categorifiable. The main
goal of this note is to provide a somewhat more direct proof of this result. In the last subsection we discuss integrality properties of the Liu-Palcoux-Wu invariants.
We classify various types of graded extensions of a finite braided tensor category $cal B$ in terms of its $2$-categorical Picard groups. In particular, we prove that braided extensions of $cal B$ by a finite group $A$ correspond to braided monoidal
$2$-functors from $A$ to the braided $2$-categorical Picard group of $cal B$ (consisting of invertible central $cal B$-module categories). Such functors can be expressed in terms of the Eilnberg-Mac~Lane cohomology. We describe in detail braided $2$-categorical Picard groups of symmetric fusion categories and of pointed braided fusion categories.
We prove that representations of the braid groups coming from weakly group-theoretical braided fusion categories have finite images.
We establish rank-finiteness for the class of $G$-crossed braided fusion categories, generalizing the recent result for modular categories and including the important case of braided fusion categories. This necessitates a study of slightly degenerate
braided fusion categories and their centers, which are interesting for their own sake.
We show that braidings on a fusion category $mathcal{C}$ correspond to certain fusion subcategories of the center of $mathcal{C}$ transversal to the canonical Lagrangian algebra. This allows to classify braidings on non-degenerate and group-theoretical fusion categories.
We classify finite pointed braided tensor categories admitting a fiber functor in terms of bilinear forms on symmetric Yetter-Drinfeld modules over abelian groups. We describe the groupoid formed by braided equivalences of such categories in terms of
certain metric data, generalizing the well-known result of Joyal and Street for fusion categories. We study symmetric centers and ribbon structures of pointed braided tensor categories and examine their Drinfeld centers.
We develop methods of computation of the Brauer-Picard groups of fusion categories and apply them to compute such groups for several classes of fusion categories of prime power dimension: representation categories of elementary abelian groups with tw
isted associativity constraint, extra special p-groups, and the Kac-Paljutkin Hopf algebra. We conclude that many finite groups of Lie type occur as composition factors of the Brauer-Picard groups of pointed fusion categories.
Let C_n denote the representation category of a finite supergroup generated by purely odd n-dimensional vector space. We compute the Brauer-Picard group BrPic(C_n) of C_n. This is done by identifying BrPic(C_n) with the group of braided tensor autoeq
uivalences of the Drinfeld center of C_n and studying the action of the latter group on the categorical Lagrangian Grassmannian of C_n. We show that this action corresponds to the action of a projective symplectic group on a classical Lagrangian Grassmannian.
We analyze the action of the Brauer-Picard group of a pointed fusion category on the set of Lagrangian subcategories of its center. Using this action we compute the Brauer-Picard groups of pointed fusion categories associated to several classical fin
ite groups. As an application, we construct new examples of weakly group-theoretical fusion categories.
Let C be a fusion category faithfully graded by a finite group G and let D be the trivial component of this grading. The center Z(C) of C is shown to be canonically equivalent to a G-equivariantization of the relative center Z_D(C). We use this resul
t to obtain a criterion for C to be group-theoretical and apply it to Tambara-Yamagami fusion categories. We also find several new series of modular categories by analyzing the centers of Tambara-Yamagami categories. Finally, we prove a general result about existence of zeroes in S-matrices of weakly integral modular categories.
