A study on duality of Hopf modules

设H是域k上的Hopf代数，A是右H-余模代数，D是A的所有余不变量的子代数，M是从A到D的所有左D-线性映射的(A，D)-双模，如果H是点Hopf代数，A是单的(A，H)-Hopf模，有限维是D-模，则通过适当扭曲A的D-模结构和M的H-余模结构，A与M同构为(A，H)-Hopf模和(A，D)-双模.这一结果可以看作是有限维Hopf代数的对偶性到Hopf模的推广。此外，设有限维Hopf代数H作用在除法代数D上，A是D#H的包含D的右H-余模子代数，则这种对偶蕴含了A中积分的一些性质。此外，设R是素代数，K是它的扩张质心，H是通过X-外作用作用于R的有限维点Hopf代数。然后，根据积分的这些性质，我们得到了包含不变子代数的R的所有有理完备子代数的集合与包括K的K#H的所有右H-余模子代数的集合之间的任意特征的一一Galois-型对应。为了进一步研究，我们有一个问题，这些结果是否可以推广到包括Hopf代数的对象的Hopf代数体。我们还有一个问题，在比上述结果更弱的条件下，是否可能给出Hopf代数作用的Galois对应定理。

Assume that H is a Hopf algebra over a field k, A a right H-comodule algebra, D the subalgebra of all coinvariants of A, and M the (A,D)-bimodule of all left D-linear maps from A to D. If H is a pointed Hopf algebra, and A is simple as an (A,H)-Hopf module and finite-dimensional as a D-module, then, by twisting the D-module structure of A and the H-comodule structure of M suitably, A is isomorphic to M as an (A,H)-Hopf module and as an (A,D)-bimodule. This result can be considered as a generalization of duality of finite-dimensional Hopf algebras to Hopf modules. Besides, suppose that a finite-dimensional Hopf algebra H acts on a division algebra D and A is a right H-comodule subalgebra of D#H including D. Then this duality implies some properties of integrals in A. Moreover, let R be a prime algebra, K its extended centreoid, and H a finite-dimensional pointed Hopf algebra acting on R by an X-outer action. Then, from those properties of integrals, we have a one to one Galois-type correspondence in arbitrary characteristic between the set of all rationally complete subalgebras of R including the subalgebra of invariants and the set of all right H-comodule subalgebras of K#H including K. For a further study, we have a problem whether these results can be generalized to Hopf algebroids, which are objects including Hopf algebras. We also have a problem whether it is possible to give a Galois correspondence theorem for Hopf algebra actions under a condition which is weaker than that of the result above.