An Approach to Quasi-hopf Algebras via Frobenius Coordinates

We study quasi-Hopf algebras and their subobjects over certain commutative rings from the point of view of Frobenius algebras. We introduce a type of Radford formula involving an anti-automorphism and the Nakayama automorphism of a Frobenius algebra, then view several results in quantum algebras from this vantage-point. In addition, separability and strong separability of quasi-Hopf algebras are studied as Frobenius algebras. Dedicated to A.A. Stolin on his fiftieth birthday In [5], Drinfel’d introduces quasi-Hopf algebras over a commutative ground ring, and works out the fundamentals of this theory of quasi-bialgebra with antipode. From a categorical point of view, modules over a quasi-bialgebra form a monoidal category with a nontrivial associativity constraint. Conjugating the comultiplication of a bialgebra by a gauge element produces nontrivial examples of quasibialgebras. Quasi-bialgebras then differ from bialgebras by being only coassociative up to conjugation by a three-cocycle; cf. Eqs. (1) and (2). Several applications are made by Drinfel’d to the Knizhnik-Zamolodchikov system of p.d.e.’s and to Reshetikhin’s method for obtaining knot invariants. Hausser and Nill have shown in [8] that finite-dimensional quasi-Hopf algebras over fields are Frobenius algebras. We would like to return in this paper to the general commutative ground ring for quasi-Hopf algebras as much as possible while retaining aspects of Frobenius algebras. In the preliminaries, we first show that quasi-Hopf algebras over a commutative ring k with trivial Picard group are Frobenius k-algebras by sketching the direct approach of Bulacu-Caenepeel [3] to the isomorphism ∫ l H ⊗H ∼= H of a quasi-Hopf algebra H , its dual H and its space of left integrals ∫ l H via the Van Daele-Panaite-Van Oystaeyen projection P : H → ∫ l H . Somewhat more generally, we introduce QFH-algebras, which are quasi-Hopf algebras over commutative rings that are Frobenius algebras. We then study a Frobenius coordinate system derived from [3], transform it to the Frobenius system introduced in Hausser-Nill [8] and make various deductions starting from a type of Radford formula for an anti-automorphism of a Frobenius algebra (Lemma 3.1). First, a simplified proof and extension of the Hausser-Nill-Radford formula for the fourth power of the antipode is provided for QFH-algebras (Theorem 3.3). Second, a quasi-Hopf subalgebra, stable under an antipode of H , is a β-Frobenius extension (Theorem 3.4). In section 2 we make a study when a quasi-Hopf algebra is separable or strongly separable in the sense of Kanzaki. 1991 Mathematics Subject Classification. 16W35, 81R05.