In this paper the authors present a systematic study of fusion categories in full generality. Let \(k\) be an algebraically closed field of characteristic zero. (In the last section of the paper some results are generalized to positive characteristic.) A fusion category over \(k\) is a \(k\)-linear semisimple rigid tensor category \(\mathcal C\) with finitely many simple objects and finite dimensional spaces of morphisms, such that the endomorphism algebra of the neutral object is \(k\) (without the last assumption, \(\mathcal C\) is called a multifusion category). Many results of the paper rely on the fact, attributed toT. Hayashi [math.QA/9904073 (1999)] andK. Szlachányi [Fields Inst. Commun. 30, 393-407 (2001;Zbl 1022.18007)] (see also the paper byV. Ostrik [Transform. Groups 8, No. 2, 177-206 (2003;Zbl 1044.18004)]) that any multi-fusion category is equivalent to the category of finite dimensional representations of a regular semisimple weak Hopf algebra.
It is shown that if \(k\) is the field of complex numbers, then the squared norm of a simple object is always positive. In particular, the global dimension of a fusion category is nonzero. The proof of this result uses that in a fusion category the identity functor is isomorphic to the functor \(****\); this is in turn a consequence of the weak Hopf algebra analogue of Radford’s formula for the fourth power of the antipode in a Hopf algebra due to the second author [Adv. Math. 170, No. 2, 257-286 (2002;Zbl 1010.16041)]. The authors conjecture that any fusion category admits a pivotal structure, that is, an isomorphism of tensor functors between the identity functor and the functor \(**\). They prove that the conjecture is true for the representation category of a finite dimensional semisimple quasi-Hopf algebra.
Some results are established for pivotal, spherical and modular categories, in particular when \(k=\mathbb{C}\), it shown that the projective representation (and indeed all its algebraic conjugates as well) of \(\text{SL}_2(\mathbb{C})\) associated to a modular category is unitary in the standard Hermitian metric.
Results on module categories and duals with respect to module categories are given. It is shown that for an indecomposable module category \(\mathcal M\) over a fusion category \(\mathcal C\), the dual category \(\mathcal C_{\mathcal M}^*\) of module functors \(\mathcal M\to\mathcal M\) is also a fusion category of the same global dimension. This had been proved byM. Müger [J. Pure Appl. Algebra 180, No. 1-2, 81-157 (2003;Zbl 1033.18002), ibid. 159-219 (2003;Zbl 1033.18003)], under the assumption that \(\dim\mathcal C\neq 0\). The paper gives also a proof of Ocneanu rigidity for multi-fusion categories: multi-fusion categories and tensor functors between them are indeformable. This extends results of Blanchard and Wassermann. As a consequence, it is shown that a module category over a multi-fusion category does not admit nontrivial deformations.
The theory of Frobenius-Perron dimensions is developed for an arbitrary fusion category and several results are proved. Some of them generalize celebrated results for finite dimensional Hopf algebras, like the Nichols-Zoeller freeness theorem and the class equation of G. I. Kac and Y.-C. Zhu. A subsection is devoted to the comparison of the global and the Frobenius-Perron dimension, where the authors discuss pseudo-unitary fusion categories, i.e., those for which the global dimension of \(\mathcal C\) equals the Frobenius-Perron dimension of \(\mathcal C\) and prove that any such fusion category admits a unique spherical structure with respect to which the categorical dimensions of simple objects coincide with the Frobenius-Perron dimensions.
Concerning the classification problem of fusion categories, the authors prove that a fusion category of prime dimension \(p\) is necessarily isomorphic to the category of representations of the group of order \(p\) with associativity constraint defined by a \(3\)-cohomology class. Fusion categories of dimension \(p^2\) are also classified. For a fusion category \(\mathcal C\) of Frobenius-Perron dimension \(p^n\) it is shown that \(\mathcal C\) admits a filtration by fusion subcategories \(\langle 1\rangle=\mathcal C^{(n)}\subseteq\cdots\subseteq\mathcal C^{(1)}\subseteq\mathcal C\), where \(\dim\mathcal C^{(i)}=p^{n-i}\). The authors discuss group-theoretical categories defined as the duals of pointed categories. In this context, they ask if every semisimple Hopf algebra is group-theoretical.

Reviewer: Sonia Natale (Cordoba)