Morita context, partial Hopf Galois extensions and partial entwining structure. (Q2341957)

Let \(k\) be a field, \(H\) a Hopf algebra with comultiplication \(\Delta\) and \(A\) a left \(H\)-module algebra over \(k\). If there exists a \(k\)-linear map \(\nearrow\colon H\otimes A\to A\) such that \(h\nearrow(ab)=\sum(h_1\nearrow a)(h_2\nearrow b)\), \(1_H\nearrow a=a\), and \(h\nearrow(g\nearrow a)=\sum(h_1\nearrow 1_A)(h_2g\nearrow a)\) where \(\Delta(h)=h_1\otimes h_2\) for all \(g,h\in H\), \(a\in A\), then \(A\) is called a left partial \(H\)-module algebra, and \(A\) is called a right partial \(H\)-comodule algebra if there exists a \(k\)-linear map \(\rho\colon A\to A\otimes H\) called a partial comodule structure map such that \((id_A\otimes\varepsilon)\rho^A=id_A\), \((\rho\otimes id_H)\rho(a)=(\rho(1_A)\otimes id_H)(id_A\otimes\Delta)\rho(a)\) and \(\rho(ab)=\rho(a)\rho(b)\) for all \(a,b\in A\) where \(\varepsilon\) is the counit of \(H\). Let \(B\) be a left partial \(H\)-comodule algebra. Then the generalized left partial smash product \(A*_l^HB\) of \(A\) and \(B\) is defined and some properties are given. Under the coaction \(\rho\) of \(H\) on \(A\), \(A^{coH}=\{a\in A\mid\rho(a)=a\rho(1_A)\}\), and the unique maximal left (right) rational submodule of the left (right) \(H^*\)-module \(H^*\) is denoted by \(H^{*rat}\). The authors construct two Morita contexts connecting \(A*H^{*rat}\) and \(A^{coH}\) for a co-Frobenius Hopf algebra \(H\) (that is, \(H\) has nonzero space of left (right) integral). Moreover, a Hopf partial \(H\)-Galois extension \(A\) of \(A^{coH}\) is defined and it is shown that \(A\) induces a unique partial entwining map compatible with the right partial coaction.