Radford's biproduct for quasi-Hopf algebras and bosonization
Let \(H\) be a Hopf algebra and \(B\) a linear space such that \(B\) is an algebra in the category of left \(H\)-modules, and a coalgebra in the category of left \(H\)-comodules. Suppose that \(B\) is a Hopf algebra in the braided category of Yetter-Drinfeld modules over \(H\). Then the smash product algebra structure and the smash coproduct coalgebra structure make \(B\otimes H\) a Hopf algebra. This is in fact a necessary and sufficient condition for the smash product algebra structure and the smash coproduct coalgebra structure to afford a Hopf algebra structure on \(B\otimes H\) and is due to Radford. Moreover, Hopf algebras in the category of Yetter-Drinfeld modules over \(H\) are in correspondence with Hopf algebras \(A\) equipped with projections \(\pi\colon A\to H\) which admit sections \(i\colon H\to A\). This correspondence when formulated in the language of braided categories is known as bosonization and is due to Majid. In the present paper, the authors generalize the above result for quasi-Hopf algebras and describe the Majid bosonization for quasi-Hopf algebras.
scientific article; zbMATH DE number 1839047