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A study on duality of Hopf modules

Hopf模对偶性的研究

基本信息

批准号：
15540054
负责人：
YANAI Tadashi
金额：
$ 0.64万
依托单位：
Niihama National College of Technology
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2004
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540054/
关键词：
Hopf algebras Hopf modules Duality Integrals Galois correspondence クロスK双代数ガロア型対応

项目摘要

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.

假设 H 是域 k 上的 Hopf 代数，A 是右 H 余模代数，D 是 A 的所有协变体的子代数，M 是从 A 到 D 的所有左 D 线性映射的 (A,D) 双模。如果 H 是指向的 Hopf 代数，并且 A 是简单的 (A,H)-Hopf 模和有限维的 D 模，然后，通过适当扭转A的D-模结构和M的H-共模结构，A与M同构为(A,H)-Hopf模和(A,D)-双模。这个结果可以被认为是有限维Hopf代数对偶性到Hopf模的推广。此外，假设有限维 Hopf 代数 H 作用于除代数 D，并且 A 是包括 D 在内的 D#H 的右 H 余模子代数。则这种对偶性隐含了 A 中积分的某些性质。此外，令 R 为素代数，K 为其扩展质心，H 为通过 X 外作用作用于 R 的有限维尖头 Hopf 代数。然后，根据积分的这些性质，我们在 R 的所有有理完备子代数集合（包括不变量子代数）和 K#H 的所有正确 H-余模子代数集合（包括 K）之间的任意特征上具有一一对应的伽罗瓦型对应关系。为了进一步研究，我们有一个问题是否可以将这些结果推广到 Hopf 代数体，这些代数是包括 Hopf 的对象代数。我们还有一个问题，是否可以在比上面结果弱的条件下给出 Hopf 代数动作的伽罗瓦对应定理。