The crossed coproduct Hopf algebras (Q2756098)

Let \(H\) be a bialgebra over a field \(k\) and let \(B\) be simultaneously an algebra and a coalgebra endowed with an action \(H\otimes B\to B\) and a coaction \(\rho\colon B\to H\otimes B\). Consider the smash product multiplication and the twisted smash coproduct comultiplication with respect to a dual cocycle \(\alpha\colon B\to H\otimes H\) on the vector space \(B\otimes H\). The author gives necessary and sufficient conditions for \(B\otimes H\) to be a bialgebra, denoted by \(B\bowtie^\alpha H\), with the above multiplication and comultiplication. In the case where \(\alpha\) is trivial, this recovers the results in \textit{D. E. Radford}'s paper [J. Algebra 92, 322-347 (1985; Zbl 0549.16003)]. Sufficient conditions are given for the existence of an antipode making \(B\bowtie^\alpha H\) into a Hopf algebra. In analogy with Radford's paper, it is shown that this construction is characterized by a weak coaction admissible mapping system. In the finite dimensional context, these results are dual to those in the paper [Commun. Algebra 26, No. 4, 1293-1303 (1998; Zbl 0898.16028)] by \textit{Z. Jiao, S. Wang} and \textit{W. Zhao}.