A study on duality of Hopf modules

假设H是一个霍普夫代数领域k,右H-comodule代数,D的所有coinvariants的子代数,和M (a, D)双模的离开维线性映射从a到D .如果H是指出霍普夫代数,和一个简单作为一个(a, H)霍普夫模块和有限维D模块,然后,由扭曲的D模块结构和H-comodule M适当,结构同构于M作为(a, H)霍普夫模块和(a, D)双模。这个结果可以看作是有限维Hopf代数对偶性在Hopf模上的推广。另外，设一个有限维Hopf代数H作用于一个除法代数D， a是d# H的一个包含D的右H模子代数，则这种对偶性暗示了a中积分的一些性质，并且设R是一个素数代数，K是它的扩展质心，H是一个有限维点Hopf代数，作用于R的是一个x外作用。然后，根据积分的这些性质，我们得到了R的包含不变量子代数的所有合理完备子代数的集合与k# H的包含K的所有正确H-模子代数的集合在任意特征上的一一伽罗瓦型对应关系。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含Hopf代数的Hopf代数群上。我们也有一个问题，是否可能在比上述结果弱的条件下给出Hopf代数作用的伽罗瓦对应定理。

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.