Quasi-quantum groups obtained from tensor braided Hopf algebras
In the present article, the author extends to the setting of quasi-Hopf algebras two very well-known facts on Hopf algebras. In the following \(H\) denotes a quasi-Hopf algebra. First, the author shows that the braiding of the category of Yetter-Drinfeld modules \({}_H^H\mathcal{YD}\) can be transported to the category of two-sided two-cosided Hopf modules \({}_H^H\mathcal{M}_H^H\) via a monoidal equivalence constructed in [\textit{D. Bulacu}, Theory Appl. Categ. 32, 1--30 (2017; Zbl 1359.16034) ] and [\textit{P. Schauenburg}, Trans. Am. Math. Soc. 354, No. 8, 3349--3378 (2002; Zbl 1001.16029) ]. In consequence, this equivalence becomes a braided monoidal equivalence. Second, the author proves that there is a bijective correspondence between quasi-Hopf maps \(H\underset{i}{\overset{\pi}{\leftrightarrows}}A\) such that \(\pi \circ i=\operatorname{id}_A\) and Hopf algebras in \({}_H^H\mathcal{YD}\). This bijection generalizes the famous construction due to Radford [\textit{D. E. Radford}, J. Algebra 92, 322--347 (1985; Zbl 0549.16003)] and it is constructed in the same way: in one direction, one must to take coinvariants, and to come back, one must to form a biproduct. As an example, the author applies the results to the quasi-Hopf algebras associated to cyclic groups.
scientific article; zbMATH DE number 7339565
scientific article