Hopf Galois extensions, smash products, and Morita equivalence (Q919084)

Let \(H\) be a finite-dimensional Hopf algebra over a field, \(A\) an \(H\)-module algebra, \(A^ H\) its ring of invariants. The authors discuss various relations between \(A^ H\), \(A\) and the smash-product \(A{\#}H\), in particular concerning \(H\) (or \(H^*\))-Galois extensions, Morita contexts and normal bases. We only mention a few which are of more general interest. \(A/A^ H\) is Galois if \(A{\#}H\) is simple (it is also assumed here that \(A\) is left irreducible over \(A{\#}H\) and has finite left Goldie rank). They show that \(A^ H\) and \(A{\#}H\) are connected via a Morita context. They also show that if \(A{\#}H\) is simple artinian, then \(A/A^ H\) has a normal basis. A major application is for \(A=D\), a division ring. They show that \((D:D^ H)=\dim D\) is equivalent to each of an extensive list of conditions, e.g. \(D/D^ H\) is an \(H^*\)-Galois extension or \(D/D^ H\) has the normal bases property. In this case, \(D{\#}H\) is isomorphic to a full matrix algebra over \(D^ H\).