Quasi-Hopf algebras and the centre of a tensor category (Q2762043)

Let \(H\) be a finite-dimensional Hopf algebra and \(D(H)\) its quantum double. Then the category of left \(H\)-modules is a tensor category and its center is braided-equivalent to the category of left \(D(H)\)-modules. The aim of this paper is to prove an analogue of this result for quasi-Hopf algebras of the form \(H^*_\omega\).NEWLINENEWLINENEWLINESpecifically, let \(H\) be a finite-dimensional cocommutative Hopf algebra and \(\omega\) a normalized 3-cocycle in the Sweedler cohomology. Since \(\omega\) is an element in \((H\otimes H\otimes H)^*\) it can be identified with an element in \(H^*\otimes H^*\otimes H^*\). The convolution inverse of \(\omega\) is a Drinfeld associator for the Hopf algebra \(H^*\) (with its usual dual structure) and is proved to give rise to a quasi-Hopf algebra structure on \(H^*\), denoted by \(H^*_\omega\). The given data give rise to another quasi-Hopf algebra constructed by Bulaco and Panaite, denoted by \(D^\omega(H)\) which has an underlying algebra of a crossed product \(H^*\#_\sigma H\). As the paper was written the authors were aware of a general construction of a quantum double for any finite-dimensional quasi-Hopf algebra. Results of this paper show that \(D^\omega(H)\) is in fact the quantum double of \(H^*_\omega\).NEWLINENEWLINENEWLINEThe main result of this paper is then that the center of the tensor category of left \(H^*_\omega\)-modules is braided equivalent to the category of left modules over the quantum double of the quasi-Hopf algebra \(H^*_\omega\).NEWLINENEWLINEFor the entire collection see [Zbl 0958.00024].