A classical result in the theory of Hopf algebras concerns the uniqueness and existence of integrals: for an arbitrary Hopf algebra, the integral space has dimension ≤ 1, and for a finite dimensional Hopf algebra, this dimension is exaclty one. We generalize these results to quasi-Hopf algebras and dual quasi-Hopf algebras. In particular, it will follow that the bijectivity of the antipode follows from the other axioms of a finite dimensional quasi-Hopf algebra. We give a new version of the Fundamental Theorem for quasi-Hopf algebras. We show that a dual quasi-Hopf algebra is co-Frobenius if and only if it has a non-zero integral. In this case, the space of left or right integrals has dimension one.
IntroductionQuasi-bialgebras and quasi-Hopf algebras were introduced by Drinfeld in [10], in connection with the Knizhnik-Zamolodchikov system of partial differential equations, cf. [14]. From a categorical point of view, the notion is not so different from classical bialgebras: we consider an algebra H, and we want to make the category of H-modules, equipped with the tensor product of vector spaces, into a monoidal category. If we require that the associativity constraint is the natural associativity condition for vector spaces, then we obtain a bialgebra structure on H, in general, we obtain a quasi-bialgebra structure, that is, we have a comultiplication and a counit on H, where the comultiplication is not necessarily coassociative, but only quasi-coassociative. Of course the theory of quasi-bialgebras and quasi-Hopf algebras is technically more complicated than the classical Hopf algebra theory. A more conceptual difference however, is the fact that the definition of a bialgebra is self-dual, and this symmetry is broken when we pass to quasi-bialgebras. As a consequence, we don't have the notion of comodule or Hopf module over a quasi-Hopf algebra, and results in Hopf algebras that depend on these notions cannot be generalized in a straightforward way. For instance, the classical proof of the uniqueness and existence of integral is based on the Fundamental Theorem for Hopf modules [24]. Hausser and Nill [13] proved that a finite dimensional quasi-Hopf algebra is a Frobenius algebra, and has a one dimensional integral space. Independently, Panaite and Van Oystaeyen [19] proved the existence of integrals for finite dimensional quasi-Hopf algebras, using the approach developed in [26], without using quasi-Hopf bimodules. * Research supported by the bilateral project "Hopf Algebras in Algebra, Topology, Geometry and Physics" of the Flemish and Romanian governments. † This paper was written while the first author was visiting the Vrije Universiteit Brussel, and he would like to thank VUB for its warm hospitality.
