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.