Using a variety of methods developed in the literature (in particular, the theory of weak Hopf algebras), we prove a number of general results about fusion categories in characteristic zero. We show that the global dimension of a fusion category is always positive, and that the S-matrix of any (not necessarily hermitian) modular category is unitary. We also show that the category of module functors between two module categories over a fusion category is semisimple, and that fusion categories and tensor functors between them are undeformable (generalized Ocneanu rigidity). In particular the number of such categories (functors) realizing a given fusion datum is finite. Finally, we develop the theory of Frobenius-Perron dimensions in an arbitrary fusion category. At the end of the paper we generalize some of these results to positive characteristic.

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On fusion categories

Using a variety of methods developed in the literature (in particular, the theory of weak Hopf algebras), we prove a number of general results about fusion categories in characteristic zero. We show that the global dimension of a fusion category is always positive, and that the S-matrix of any modular category (not necessarily hermitian) is unitary. We also show that the category of module functors between two module categories over a fusion category is semisimple, and that fusion categories and tensor functors between them are undeformable (generalized Ocneanu rigidity). In particular the number of such categories (functors) realizing a given fusion datum is finite. Finally, we develop the theory of Frobenius-Perron dimensions in an arbitrary fusion category and classify categories of prime dimension.

We introduce quasi-Hopf $*$-algebras ie quasi-Hopf algebras equipped with a conjugation (star) operation The definition of quasi-Hopf $*$-algebras proposed ensures that the class of quasi-Hopf $*$-algebras is closed under twisting and additionally, that any Hopf $*$-algebra becomes a quasi-Hopf $*$-algebra via twisting The basic properties of these algebras are developed The relationship between the antipode and star structure is investigated Quasi-triangular quasi-Hopf $*$-algebras are introduced and studied

Quasi-Hopf $*$-Algebras

A two-sided coaction of a Hopf algebra on an associative algebra ℳ is an algebra map of the form , where (λ,ρ) is a commuting pair of left and right -coactions on ℳ, respectively. Denoting the associated commuting right and left actions of the dual Hopf algebra on ℳ by ◃ and ▹, respectively, we define the diagonal crossed product to be the algebra generated by ℳ and with relations given by We give a natural generalization of this construction to the case where is a quasi-Hopf algebra in the sense of Drinfeld and, more generally, also in the sense of Mack and Schomerus (i.e. where the coproduct Δ is non-unital). In these cases our diagonal crossed product will still be an associative algebra structure on extending , even though the analogue of an ordinary crossed product in general is not well defined as an associative algebra. Applications of our formalism include the field algebra constructions with quasi-quantum group symmetry given by G. Mack and V. Schomerus [31, 47] as well as the formulation of Hopf spin chains or lattice current algebras based on truncated quantum groups at roots of unity. In the case and λ=ρ=Δ we obtain an explicit definition of the quantum double for quasi-Hopf algebras , which before had been described in the form of an implicit Tannaka–Krein reconstruction procedure by S. Majid [35]. We prove that is itself a (weak) quasi-bialgebra and that any diagonal crossed product naturally admits a two-sided -coaction. In particular, the above-mentioned lattice models always admit the quantum double as a localized cosymmetry, generalizing results of Nill and Szlachanyi [42]. A complete proof that is even a (weak) quasi-triangular quasi-Hopf algebra will be given in a separate paper [27].

Diagonal crossed products by duals of quasi-quantum groups

We develop the theory of Hopf bimodules for a finite rigid tensor category C. Then we use this theory to define a distinguished invertible object D of C and an isomorphism of tensor functors δ:V**→D⊗**V⊗D−1. This provides a categorical generalization of Radford's S 4 formula for finite-dimensional Hopf algebras (presented in 1976), which was proved by Nikshych (2002) for weak Hopf algebras and by Hausser and Nill (1999) for quasi-Hopf algebras, and was conjectured in general by Etingof and Ostrik (2003). When C is braided, we establish a connection between δ and the Drinfeld isomorphism of C, extending the result of Radford (1992). We also show that a factorizable braided tensor category is unimodular (i.e., D=1). Finally, we apply our theory to prove that the pivotalization of a fusion category is spherical, and give a purely algebraic characterization of exact module categories defined by Etingof and Ostrik (2003).

An analogue of Radford's S4 formula for finite tensor categories

We generalize the fundamental structure Theorem on Hopf (bi)-modules by Larson and Sweedler to quasi-Hopf algebras H. If H is finite dimensional this proves the existence and uniqueness (up to scalar multiples) of integrals in H. Among other applications we prove a Maschke type Theorem for diagonal crossed products as constructed by the authors.

Integral Theory for Quasi-Hopf Algebras

In [Dr1] Drinfeld showed that any finite dimensional Hopf algebra  extends to a quasitriangular Hopf algebra , the quantum double of . Based on the construction of a so-called diagonal crossed product developed by the authors in [HN], we generalize this result to the case of quasi-Hopf algebras . As for ordinary Hopf algebras, as a vector space the “quasi-quantum double” is isomorphic to , where  denotes the dual of . We give explicit formulas for the product, the coproduct, the R-matrix and the antipode on  and prove that they fulfill Drinfeld's axioms of a quasitriangular quasi-Hopf algebra. In particular  becomes an associative algebra containing  as a quasi-Hopf subalgebra. On the other hand,  is not a subalgebra of  unless the coproduct on  is strictly coassociative. It is shown that the category  of finite dimensional representations of  coincides with what has been called the double category of -modules by S. Majid [M2]. Thus our construction gives a concrete realization of Majid's abstract definition of quasi-quantum doubles in terms of a Tannaka–Krein-like reconstruction procedure. The whole construction is shown to generalize to weak quasi-Hopf algebras with  now being linearly isomorphic to a subspace of .

Doubles of Quasi-Quantum Groups

Drinfeld showed that any finite dimensional Hopf algebra \G extends to a quasitriangular Hopf algebra \D(\G), the quantum double of \G. Based on the construction of a so--called diagonal crossed product developed by the authors, we generalize this result to the case of quasi--Hopf algebras \G. As for ordinary Hopf algebras, as a vector space the ``quasi--quantum double'' \D(\G) is isomorphic to the tensor product of \G and its dual \dG. We give explicit formulas for the product, the coproduct, the R--matrix and the antipode on \D(\G) and prove that they fulfill Drinfeld's axioms of a quasitriangular quasi--Hopf algebra. In particular \D(\G) becomes an associative algebra containing \G as a quasi--Hopf subalgebra. On the other hand, \dG \otimes 1 is not a subalgebra of \D(\G) unless the coproduct on \G is strictly coassociative. It is shown that the category of finite dimensional representations of \D(\G) coincides with what has been called the double category of \G--modules by S. Majid [M2]. Thus our construction gives a concrete realization of Majid's abstract definition of quasi--quantum doubles in terms of a Tannaka--Krein--like reconstruction procedure. The whole construction is shown to generalize to weak quasi--Hopf algebras with \D(\G) now being linearly isomorphic to a subspace of \dG \otimes \G.

Let \G be a (weak) quasi-Hopf algebra. Using a two-sided \G-coaction on an algebra \M, we construct what we call the diagonal crossed product as a new associative algebra structure on \M\otimes \dG, where \dG is the dual of \G. This construction is largely motivated by the special case \M = \G, for which we obtain an explicit definition of the quantum double \D(\G) for quasi-Hopf algebras. Applications of our formalism include the field algebra construction of Mack and Schomerus as well as the formulation of Hopf Spin chains or lattice current algebras based on truncated quantum groups at roots of unity. A complete proof that \D(\G) is even a (weak) quasi-triangular quasi-Hopf algebra will be given in a separate paper.

Frank Hausser

Papers

Diagonal crossed products by duals of quasi-quantum groups

Integral Theory for Quasi-Hopf Algebras

Doubles of Quasi-Quantum Groups

Doubles of Quasi-Quantum Groups

Diagonal Crossed Products by Duals of Quasi-Quantum Groups