A study on new developments of Hopf-Galois correspondence

设K是域，A是X-K Hopf代数。设A是有限维的左K-空间，指向的是K-余代数。设R是对称商代数Q的中心为K的素代数，则A作用于Q上的作用是连续的且外作用的.如果A有一个非零左积分，则R的一个含有R的H-不变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。若R在A的任一群元作用下稳定，则该对应映射是内射的.设A^*是A到K的所有右K-映射的集合，B是A的右余理想子代数.如果B有一个非零的左积分t且t生成B为左A*-模，则上述对应映射是满射的.其次设R是域k上的素代数，H是作用于R上的有外作用的有限维点Hopf代数.在H是群代数的情况下，研究了下列两个问题，并给出了肯定的回答。设U是R的一个包含H-不变子代数RH的子代数。2.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的商Hopf代数，则H’在R的I不变子代数上的作用是外的吗？

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?