Morita, Galois, and the trace map: a survey

Abstract

Let A be a \({\mathcal {K}}\)-algebra and H a \({\mathcal {K}}\)-bialgebra (\({\mathcal {K}}\) being a field). Any action \(\beta \) of H on A gives rise to two new \({\mathcal {K}}\)-algebras, namely, the algebra \(A^\beta \) of the invariants of A under \(\beta \) and the smash product \(A\#_\beta H\), as well as a canonical Morita context connecting them. Such a context keeps a close relation with the notion of Galois extension. Indeed, in some cases where it makes sense the strictness of this context is equivalent to exactly say that A is a \(H^*\)-Galois extension of \(A^\beta \). In general, such an equivalence depends also on the surjectivity of a certain trace map from A to \(A^\beta \). This paper is a survey about the strictness of this context in the setting of partial actions of groups and of Hopf algebras.