Nilpotent fusion categories

By Shlomo Gelaki, D. Nikshych. 2006

178 citations recorded by the source corpus.

Abstract In this paper we extend categorically the notion of a finite nilpotent group to fusion categories. To this end, we first analyze the trivial component of the universal grading of a fusion category C , and then introduce the upper central series of C . For fusion categories with commutative Grothendieck rings (e.g., braided fusion categories) we also introduce the lower central series. We study arithmetic and structural properties of nilpotent fusion categories, and apply our theory to modular categories and to semisimple Hopf algebras. In particular, we show that in the modular case the two central series are centralizers of each other in the sense of M. Muger.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c07af2d5f0d47bc4dc57fe9e784025ee5e649f29

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Fusion categories and homotopy theory (2009, unknown): “ a normal subgroup, and we will denote the quotient of Aut⊗(D) by im(Con) by Out⊗(D). 11.3. The group Inv(Z(D)). We will now describe the group T = Inv(Z(D)). This is a special case of Theorem 5.2 of [GN], where the group of invertible objects of a general group-theoretical category was described. A”
On braided fusion categories I (2009, methodology): “18(iii) and using the fact that any finite subgroup of Q>0/(Q × >0) 2 is a 2-group one gets the following statement from [22].”
A Finiteness Property for Braided Fusion Categories (2009, background): “-hard (where the last dichotomy assumes P 6= NP ).”
Hierarchy of Topological Order From Finite-Depth Unitaries, Measurement, and Feedforward (2022, unknown): “Namely, we conjecture that all nil-2 MTCs(18) [112] can be prepared in one shot using quasi-FDLUs.”
GROUP-THEORETICAL PROPERTIES OF NILPOTENT MODULAR CATEGORIES (2007, background): “TIES OF NILPOTENT MODULAR CATEGORIES 5 is called the upper central series of C. We say that a fusion category C is nilpotent if every non-trivial subcategory of C has a non-trivial group grading, see [GN]. Equivalently, C is nilpotent if its upper central series converges to Vec (the category of ﬁni”