Non-degeneracy conditions for braided finite tensor categories

Kenichi Shimizu

doi:10.1016/j.aim.2019.106778

Abstract

For a braided finite tensor category C with unit object 1∈C, Lyubashenko considered a certain Hopf algebra F∈C endowed with a Hopf pairing ω:F⊗F→1 to define the notion of a ‘non-semisimple’ modular tensor category. We say that C is non-degenerate if the Hopf pairing ω is non-degenerate. In this paper, we show that C is non-degenerate if and only if it is factorizable in the sense of Etingof, Nikshych and Ostrik, if and only if its Müger center is trivial, if and only if the linear map HomC(1,F)→HomC(F,1) induced by the pairing ω is invertible. As an application, we prove that the category of Yetter-Drinfeld modules over a Hopf algebra in C is non-degenerate if and only if C is.

Highlights

The S-matrix of a ribbon fusion category C is the square matrix whose (i, j)-th entry is the invariant of the Hopf link colored with i and j, where i and j run over the isomorphism classes of simple objects of C
Is a Hopf algebra in C and has the Hopf pairing ωC : F ⊗ F → 1⁄2 defined in terms of the braiding of C
We introduce the following notion: For a full subcategory D of C, the Muger centralizer [Mug03] of D in C, denoted by D′, is defined to be the full subcategory of C consisting of all objects X ∈ C such that σY,X ◦ σX,Y = idX⊗Y for all Y ∈ D, where σ is the braiding of C

Summary

Introduction

The S-matrix of a ribbon fusion category C is the square matrix whose (i, j)-th entry is the invariant of the Hopf link colored with i and j, where i and j run over the isomorphism classes of simple objects of C. We recall that the non-degeneracy of a braided finite tensor category C is a generalization of the factorizability of a Hopf algebra [RSTS88]; see [KL01, §7.4.6] for the detail. The present paper is organized as follows: In Section 2, we fix conventions and recall basic results on finite tensor categories, Frobenius-Perron dimensions, and Hopf monads from [ML98, Kas, EGNO15, BV07, BV12, BLV11]. By using the central Hopf monad and its relation to the induction to the Drinfeld center, we prove the following formula for the Frobenius-Perron dimensions: If A and B are tensor full subcategories of C (see Definition 4.3 for the precise meaning), we have. As an application of the integral theory for unimodular finite tensor categories developed in [Shi15], we prove that (1.2) is equivalent to that the Muger center of C is trivial.

Preliminaries

Non-degeneracy and factorizability

Factorizability and the Muger center

Weak-factorizability

Non-degeneracy of the Yetter-Drinfeld category