On the structure of the \(B\)-cocleft module coalgebra (Q2716369)

Let \(H\) be a Hopf algebra which coacts weakly on a coalgebra via \(C\ni c\mapsto\sum c^{(1)}\otimes c^{(2)}\in H\otimes C\), and let \(D\) be a left \(H\)-module coalgebra via \(H\otimes D\ni h\otimes d\mapsto h\rightharpoonup d\in D\). For any \(\alpha\in\Hom(C,H\otimes H)\), denote \(\alpha(c)=\sum\alpha_1(c)\otimes\alpha_2(c)\), and define \(\overline\Delta\colon C\otimes D\to(C\otimes D)\otimes(C\otimes D)\) and \(\overline\varepsilon\colon C\otimes D\to k\) by \(\overline\Delta(c\otimes d)=\sum c_{(1)}\otimes c^{(1)}_{(2)}\alpha_1(c_{(3)})\rightharpoonup d_{(1)}\otimes c^{(2)}_{(2)}\otimes\alpha_2(c_{(3)})\rightharpoonup d_{(2)}\) and \(\overline\varepsilon(c\otimes d)=\varepsilon(c)\varepsilon(d)\). Denote \((C\otimes D,\overline\Delta,\overline\varepsilon)\) by \(C_H\times^L_\alpha D\), and \(C_\alpha[D]=C_H\times^L_\alpha D\) when the coaction on \(C\) is trivial. This paper describes an equivalent condition for \(C_H\times^L_\alpha D\) to be a coalgebra. For \(\alpha,\beta\in\Hom(C,H\otimes H)\), if \(\alpha(c)=\sum(\mu^{- 1}(c_{(2)})\otimes\mu^{-1}(c_{(1)}))\beta(c_{(3)})\Delta\mu(c_{(4)})\) for some convolution-invertible \(\mu\in\Hom(C,H)\), then \(C_H\times^L_\beta D\cong C_\alpha[D]\) as coalgebras. Then the authors discuss the right version of the above results, and also give a sufficient condition for a right \(B\)-cocleft right \(H\)-module coalgebra \(C\) to be isomorphic to \(\overline C\otimes H\).