[Paper Review] Central Exact Sequences of Tensor Categories, Equivariantization and Applications

Alain Bruguières, Sonia Natale|arXiv (Cornell University)|Dec 14, 2011

Algebraic structures and combinatorial models13 references4 citations

TL;DR

This paper introduces central exact sequences of tensor categories and establishes necessary and sufficient conditions for a tensor category exact sequence to arise from equivariantization under finite group or finite group scheme actions. It provides a characterization of equivariantizations for fusion and finite tensor categories, showing that a dominant tensor functor of Frobenius-Perron index 2 or the smallest prime factor of the Frobenius-Perron dimension of a weakly integral category is always an equivariantization.

ABSTRACT

We define equivariantization of tensor categories under tensor group scheme actions and give necessary and sufficient conditions for an exact sequence of tensor categories to be an equivariantization under a finite group or finite group scheme action. We introduce the notion of central exact sequence of tensor categories and use it in order to present an alternative formulation of some known characterizations of equivariantizations for fusion categories, and to extend these characterizations to equivariantizations of finite tensor categories under finite group scheme actions. In particular, we obtain a simple characterization of equivariantizations under actions of finite abelian groups. As an application, we show that if $\C$ is a fusion category and $F: \C o \D$ is a dominant tensor functor of Frobenius-Perron index $p$, then $F$ is an equivariantization if $p=2$, or if $\C$ is weakly integral and $p$ is the smallest prime factor of $\FPdim \C$.

Motivation & Objective

• To define and study equivariantization of tensor categories under tensor group scheme actions.
• To introduce the notion of central exact sequences of tensor categories as a tool for characterizing equivariantizations.
• To extend known characterizations of equivariantizations in fusion categories to finite tensor categories under finite group scheme actions.
• To provide a simple criterion for equivariantization when the acting group is finite and abelian.
• To establish conditions under which a dominant tensor functor between fusion categories is necessarily an equivariantization.

Proposed method

• Introduce central exact sequences of tensor categories as a generalization of exact sequences in Hopf algebra theory.
• Use the monadic approach to relate exact sequences to Hopf monads on tensor categories.
• Characterize equivariantizations via the structure of normal, faithful, and exact Hopf monads.
• Apply the theory to fusion categories and finite tensor categories, particularly under actions of finite abelian groups.
• Use Frobenius-Perron dimension and properties of the center of a category to analyze the structure of tensor functors.
• Leverage results on quasi-fiber functors and integrality to deduce conditions under which a tensor functor must be an equivariantization.

Experimental results

Research questions

• RQ1 When is an exact sequence of tensor categories an equivariantization under a finite group or finite group scheme action?
• RQ2 What characterizes a central exact sequence in the context of tensor categories?
• RQ3 Under what conditions is a dominant tensor functor of Frobenius-Perron index 2 necessarily an equivariantization?
• RQ4 Can the smallest prime factor of the Frobenius-Perron dimension of a weakly integral fusion category guarantee that a dominant functor is an equivariantization?
• RQ5 Is every fusion subcategory of index 2 in a fusion category necessarily normal?

Key findings

• A dominant tensor functor between fusion categories of Frobenius-Perron index 2 is always an equivariantization.
• For a weakly integral fusion category, if a dominant tensor functor has Frobenius-Perron index equal to the smallest prime factor of the category's Frobenius-Perron dimension, then it is an equivariantization.
• The center of a fusion category contains only invertible objects when the Frobenius-Perron dimension of a simple object is less than the smallest prime factor of the category's dimension.
• A central exact sequence of fusion categories arises from an equivariantization if and only if the associated Hopf monad is normal and faithful.
• There exist Tambara-Yamagami categories where the pointed subcategory has index 2 but is not normal, showing that index 2 does not imply normality in general.
• A Tambara-Yamagami category with Picard group of prime order is simple, as it admits no nontrivial exact sequences of fusion categories.