A generalization of crossed products (Q2708912)

The authors study a new type of crossed product. Given a Hopf algebra \(H\) and an algebra \(A\), they introduce the notion of a ``transposition'' \(s\colon H\otimes A\to A\otimes H\), which is related to ``entwined structures'' [see \textit{T. Brzeziński} and \textit{S. Majid}, Commun. Math. Phys. 191, No. 2, 467-492 (1998; Zbl 0899.55016)]. For such \(H\), \(A\) and \(s\), an associative unital multiplication on \(A\otimes H\) is defined, starting from a ``weak \(s\)-action'' \(\rho\colon H\otimes A\to A\) and a normalized cocycle \(\sigma\colon H\otimes H\to A\). This algebra structure is a particular case of the one constructed by \textit{T. Brzeziński} [Commun. Algebra 25, No. 11, 3551-3575 (1997; Zbl 0887.16026)], but covers several known examples, like Ore extensions without derivations, the crossed products of \textit{R. Blattner, M. Cohen} and \textit{S. Montgomery} [Trans. Am. Math. Soc. 298, 671-711 (1986; Zbl 0619.16004)], etc. A Maschke-type theorem for this kind of crossed products is proved, and several notions and results from \textit{R. Blattner, M. Cohen} and \textit{S. Montgomery} [loc. cit.] are extended and adapted to the present setting.NEWLINENEWLINEFor the entire collection see [Zbl 0955.00038].