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| On Hopf Galois Extension of Separable Algebras |
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Citation： |
Yu LU,Shenglin ZHU.On Hopf Galois Extension of Separable Algebras[J].Chinese Annals of Mathematics B,2017,38(4):999~1018 |
| Page view： 2840
Net amount： 1189 |
Authors： |
Yu LU; Shenglin ZHU |
Foundation： |
This work was supported by the National Natural Science
Foundation of China (No.11331006). |
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| Abstract： |
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 $A/A^H$ a right
$H^*$-Galois extension. The authors prove that, if $A^H$ is a
separable $k$-algebra, then for any right coideal subalgebra $B$ of
$H$, the $B$-invariants $A^B=\{a\in A \mid b\cdot
a=\varepsilon(b)a,\ \forall b\in B\}$ is a separable $k$-algebra.
They also establish a Galois connection between right coideal
subalgebras of $H$ and separable subalgebras of $A$ containing $A^H$
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. |
Keywords： |
Semisimple Hopf algebra, Hopf Galois extension, Separable algebra, Galois connection |
Classification： |
17B40, 17B50 |
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