While quasi-Hopf algebras were introduced by V. G. Drinfel'd (cf. ), the first authors to contemplate the notion of a ribbon quasi-Hopf algebra were D. Altschüler and A. Coste (cf. , Par. 4.1, p. 89). They define them as quasitriangular quasi-Hopf algebras with an additional central element, the ribbon element, that is subject to four axioms. However, as the authors point out themselves, these axioms are not completely satisfactory, as they neither reduce directly to the axioms of a ribbon Hopf algebra, in the case where the quasi-Hopf algebra happens to be an ordinary Hopf algebra, nor are in complete analogy to the axioms for a ribbon category. They therefore analyzed their notion further and explained that, in the case where the evaluation element α is invertible, their axioms are equivalent to a set of four different axioms which are considerably closer to the notion of a ribbon Hopf algebra and the notion of a ribbon category.
However, in the case of ribbon Hopf algebras, one of the four axioms is actually a consequence of the remaining axioms. Therefore D. Bulacu, F. Panaite, and F. van Oystaeyen proposed a different definition of a ribbon quasi-Hopf algebra, leaving out this supposedly superfluous axiom
(cf. , Def. 2.3, p. 6106). Again in the case where the evaluation element is invertible, they showed that this axiom really was superfluous, so that their definition was equivalent to the revised version of Altschüler and Coste (cf. , Prop. 5.5, p. 6119).
Of course, this raised the question whether the assumption on the invertibility of the evaluation element is really necessary to establish
these two equivalences, or whether this assumption was only made to simplify the argument. In the case of the first equivalence, between the two versions of the definition already proposed by Altschüler and Coste, this question was addressed by D. Bulacu and E. Nauwelaerts, who showed that the assumption is not necessary (cf. , Thm. 3.1, p. 667). In a recent article, when using ribbon quasi-Hopf algebras to exemplify certain properties of modular data, the authors have claimed that this assumption is also not necessary for the second equivalence between the definition of Altschüler and Coste and the definition of Bulacu, Panaite, and van Oystaeyen (cf. , Cor. 5.1, p. 50). The purpose of the present article is to prove this claim.
To do this, we take a certain viewpoint, which is suitable not only for this proof, but also for similar questions: The R-matrix can be viewed
as a twist that takes the coproduct into the coopposite coproduct. However, while twisting leaves the antipode unchanged, the
coopposite coproduct naturally comes endowed with the inverse antipode.
The so-called Drinfel'd element now appears as the element that connects these two choices for the antipode of the coopposite quasi-Hopf algebra.
Viewing the Drinfel'd element in this way enables us not only to give
a relatively easy proof of our claim, but also allows us to give a new derivation of the fundamental properties of the Drinfel'd element in a comparatively short and conceptual way.
The article consists of two sections. The first, preliminary section contains a brief summary of the basic facts about quasi-bialgebras,
quasi-Hopf algebras, quasitriangularity, and twisting. However, we trace
more precisely than the available references how some elements already introduced in Drinfel'd's original article transform under twisting and
other modifications, as this turns out to be crucial for our treatment.
The second section contains our main result, Theorem 2.3. As explained above, we prove it by viewing the R-matrix as a twist, a viewpoint developed in Paragraph 2.1. The new proof of the fundamental properties of the Drinfel'd element also mentioned above is given in Paragraph 2.2. The article concludes with Proposition 2.4, a formula for the image of the Drinfel'd element under the antipode. Although this formula was needed in our earlier proofs of Theorem 2.3, it is not needed in the proof presented here. We include it nonetheless, because it is of independent interest and its proof nicely illustrates the ideas that we have developed.
In the following, we work over a base field that is denoted by K.
All vector spaces that we will consider will be defined over this base field K, and all tensor products will be taken over K. With respect to enumeration, we use the convention that propositions, definitions, and similar items are referenced by the paragraph in which they occur; an
additional third digit indicates a part of the corresponding item.
For example, a reference to Proposition 2.2.3 refers to the third assertion of the unique proposition in Paragraph 2.2.