A finiteness property for braided fusion categories. (English) Zbl 1253.18005
In this paper the authors consider a finiteness condition, called property F, on a braided fusion category \(\mathcal C\). Having property F means that all braid group representations on the endomorphism spaces of \(n\)th tensor powers of objects of \(\mathcal C\), arising from the braiding of \(\mathcal C\), factor over finite groups. For instance, in the paper [Pac. J. Math. 234, No. 1, 33–41 (2008; Zbl 1207.16038)], P. Etingof, E. Rowell and S. Witherspoon have established that if \(\mathcal C\) is a braided group-theoretical fusion category, as in Section 8.8 of the paper [Ann. Math. (2) 162, No. 2, 581–642 (2005; Zbl 1125.16025)] by P. Etingof, D. Nikshych and V. Ostrik, then \(\mathcal C\) has property F. The authors conjecture that having property F is equivalent to \(\mathcal C\) being weakly integral, that is, having integer Frobenius-Perron dimension. Examples of fusion categories of quantum group type are presented to support this conjecture. One of the main results of the paper gives a list of sufficient conditions for a braided integral (that is, such that all Frobenius-Perron dimensions of simple objects are integers) fusion category to have property F. As part of the proof of this result, some classification results for certain classes of fusion categories are obtained: this includes fusion categories generated by a self-dual object of dimension 2 and whose simple objects have dimension 1 or 2, and also modular categories of dimension \(pq^2\) and \(pq^3\), where \(p\) and \(q\) are distinct prime numbers.
Reviewer: Sonia Natale (Córdoba)
MSC:
| 18D10 | Monoidal, symmetric monoidal and braided categories (MSC2010) |
| 20F36 | Braid groups; Artin groups |
| 16T05 | Hopf algebras and their applications |
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