Doi-Koppinen Hopf modules versus entwined modules (Q1591983)

Let \(k\) be a field. A Doi-Koppinen datum is a triple \((A,C,H)\), where \(H\) is a bialgebra over \(k\), \(A\) is a right \(H\)-comodule algebra and \(C\) is a right \(H\)-module coalgebra. A Doi-Koppinen Hopf module with respect to the Doi-Koppinen datum \((A,C,H)\) is a right \(A\)-module and a right \(C\)-comodule, such that the \(A\)-module structure map is \(C\)-colinear or, equivalently, the \(C\)-comodule structure map is \(A\)-linear. These notions have been introduced, independently by \textit{Y. Doi} [in J. Algebra 153, No. 2, 373-385 (1992; Zbl 0782.16025)] and \textit{M. Koppinen} [in J. Pure Appl. Algebra 104, No. 1, 61-80 (1995; Zbl 0838.16035)]. An entwining structure, as introduced by \textit{T. Brzeziński} [in J. Algebra 215, No. 1, 290-317 (1999; Zbl 0936.16030)], is a triple \((A,C,\psi)\) for a certain map \(\psi\colon C\otimes A\to A\otimes C\), and an entwined module with respect to the entwining structure is a right \(A\)-module and a right \(C\)-comodule, both structures satisfying a certain restriction with respect to \(\psi\). Every Doi-Koppinen datum induces an entwining structure, in such a way that the entwined modules with respect to the resulting entwining structure are the Doi-Koppinen modules with respect to the original Doi-Koppinen datum. It has been shown by \textit{D. Tambara} [J. Fac. Sci., Univ. Tokyo, Sect. I A 37, No. 2, 425-456 (1990; Zbl 0717.16030)] that when the algebra \(A\) is finite dimensional, every entwining structure can be obtained from a Doi-Koppinen datum in the above fashion. This paper provides examples which show that this is not true in general, even when \(C\) is finite dimensional.