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On Hopf Galois Hirata extensions. (English) Zbl 1034.16041

Summary: Let \(H\) be a finite-dimensional Hopf algebra over a field \(k\), \(H^*\) the dual Hopf algebra of \(H\), and \(B\) a right \(H^*\)-Galois and Hirata separable extension of \(B^H\). Then \(B\) is characterized in terms of the commutator subring \(V_B(B^H)\) of \(B^H\) in \(B\) and the smash product \(V_B(B^H)\#H\). A sufficient condition is also given for \(B\) to be an \(H^*\)-Galois Azumaya extension of \(B^H\).

MSC:

16W30 Hopf algebras (associative rings and algebras) (MSC2000)
16H05 Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.)
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