A study on new developments of Hopf-Galois correspondence

设K是域，A是X-KHopf代数.设A是有限维的左K-空间，并且是K-余代数。设R是以K为对称商代数Q的中心的素代数，则A作为右K-空间的维数与A作为左K-空间的维数一致。假设A以连续的外作用作用于Q。若A有一个非零左积分，则R的一个包含R的H-不变量子代数的有理完备子代数通过一定的对应映射对应于A的一个右上理想子代数。设A^* 是从A到K的所有右K-映射的集合，B是A的右余理想子代数，若R在A的任意类群元素的作用下稳定，则该对应映射是单射的。若B有一个非零左积分t，且t生成B为一个左A*-模，则上述对应映射是满射的。对H为群代数时的两个问题进行了研究，得到了肯定的答案.设U是R的包含H-不变量RH的子代数的子代数。U的一个固定RH的自同构能扩张到R的一个自同构吗？2.假设R的对称商代数的中心与k重合。设I是H的正规右余理想子代数，H'是H与I的商Hopf代数。那么H'在R的I-不变量的子代数上的作用是外作用吗？

Let K be a field and A a X-K Hopf algebra. Assume that A is finite-dimensional as a left K-space and pointed as a K-coalgebra. Then the dimension of A as a right K-space coincides with the dimension as a left K-space.Let R be a prime algebra having K as the center of the symmetric quotient algebra Q. Suppose that A acts on Q with continuous and outer action. If A has a nonzero left integral, then a rationally complete subalgebra of R containing the subalgebra of H-invariants of R corresponds to a right coideal subalgebra of A by a certain correspondence map. If R is stable under the action of any grouplike element of A, this correspondence map is injective.Let A^* be the set of all right K-maps from A to K and B a right coideal subalgebra of A. If B has a nonzero left integral t and t generates B as a left A*-module, then the above correspondence map is surjective.Next let R be a prime algebra over a field k and H a finite-dimensional pointed Hopf algebra acting on R with an outer action. The following two problems, which extend the results when H is a group algebra, were examined in the case of a certain example and positive answers were obtained.1. Let U be a subalgebra of R containing the subalgebra of H-invariants RH. Does an automorphism of U which fixes RH extend to an automorphism of R?2. Assume that the center of the symmetric quotient algebra of R coincides with k. Let I be a normal right coideal subalgebra of H and H' the quotient Hopf algebra of H by I. Then is the action of H' on the subalgebra of I-invariants of R outer?