Hopf algebra structure over crossed coproducts (Q1586843)

This is a difficult to read paper, due to the awkward phrasing of the text as well as the numerous spelling/typographical errors. Mathematically, the issue discussed here is: when does a crossed coproduct coalgebra inherit a Hopf algebra structure? The notation \(B\times_\alpha H\) represents the crossed coproduct coalgebra, where \(B\) is a left weakly \(H\)-comodule coalgebra and \(\alpha\) is a linear map \(B\to H\otimes H\) which is convolution invertible and a \(2\)-cocycle. Through a sequence of computation lemmas (with proofs omitted), the authors show that \(B\times_\alpha H\) is a Hopf algebra if and only if nine different conditions are satisfied. The proof of the theorem, given in the paper, is a two page calculation. As a corollary, the Heisenberg double and smash product \(H\times H^\ast\) (as defined by \textit{J.-H. Lu} [Duke Math. J. 74, No. 3, 763-776 (1994; Zbl 0815.16020)]) is shown to not inherit a Hopf algebra structure. Finally, the authors introduce the concept of an \((H,\alpha)\)-comodule, and characterize \(B_{\times_\alpha}^\#H\) (a Hopf algebra which is not defined in the paper) as such a comodule.