On Hopf Galois Extension of Separable Algebras

In this paper, the classical Galois theory to the H＊-Galois case is developed. Let H be a semisimple and cosemisimple Hopf algebra over a field k, A a left H-module algebra, and A/An a right H＊-Galois extension. The authors prove that, if An is a separable kalgebra, then for any right coideal subalg...

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Bibliographic Details
Published inChinese annals of mathematics. Serie B Vol. 38; no. 4; pp. 999 - 1018
Main Authors Lu, Yu, Zhu, Shenglin
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 01.07.2017
Springer Nature B.V
School of Mathematical Sciences,Fudan University,Shanghai 200433,China
Subjects
Algebra
Applications of Mathematics
Galois扩张
H-模代数
Hopf
Mathematics
Mathematics and Statistics
一一对应
代数域
伽罗瓦理论
可分代数
理想子代数
Semisimple Hopf algebra
Separable algebra
17B50
17B40
Galois connection
Hopf Galois extension
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Summary:In this paper, the classical Galois theory to the H＊-Galois case is developed. Let H be a semisimple and cosemisimple Hopf algebra over a field k, A a left H-module algebra, and A/An a right H＊-Galois extension. The authors prove that, if An is a separable kalgebra, then for any right coideal subalgebra B of H, the B-invariants AB = {a ∈ A ｜ b · a = ε（b）a, Ab ε B} is a separable k-algebra. They also establish a Galois connection between right coideal subalgebras of H and separable subalgebras of A containing AH as in the classical case. The results are applied to the case H = （kG）＊ for a finite group G to get a Galois 1-1 correspondence.
Bibliography:In this paper, the classical Galois theory to the H＊-Galois case is developed. Let H be a semisimple and cosemisimple Hopf algebra over a field k, A a left H-module algebra, and A/An a right H＊-Galois extension. The authors prove that, if An is a separable kalgebra, then for any right coideal subalgebra B of H, the B-invariants AB = {a ∈ A ｜ b · a = ε（b）a, Ab ε B} is a separable k-algebra. They also establish a Galois connection between right coideal subalgebras of H and separable subalgebras of A containing AH as in the classical case. The results are applied to the case H = （kG）＊ for a finite group G to get a Galois 1-1 correspondence.
31-1329/O1
Semisimple Hopf algebra, Hopf Galois extension, Separable algebra,Galois connection
ISSN:0252-9599
1860-6261
DOI:10.1007/s11401-017-1108-3