On noncommutative equivariant bundles

We discuss a possible noncommutative generalization of the notion of an equivariant vector bundle. Let $A$ be a $mathbb{K}$-algebra, $M$ a left $A$-module, $H$ a Hopf $mathbb{K}$-algebra, $delta:A to H otimes A:=H otimes_mathbb{K} A$ an algebra coaction, and let $(H otimes A)_delta$ denote $H otimes A$ with the right $A$-module structure induced by $delta$. The usual definitions of equivariant vector bundle naturally lead, in the context of $mathbb{K}$-algebras, to an $(Hotimes A)$-module homomorphism [Theta: H otimes M to (H otimes A)_delta otimes_AM] that fulfills some appropriate conditions. On the other hand, sometimes an $(A,H)$-Hopf module is considered instead, for the same purpose. When $Theta$ is invertible, as is always the case when $H$ is commutative, the two descriptions are equivalent. We point out that the two notions differ in general, by giving an example of a noncommutative Hopf algebra $H$ for which there exists such a $Theta$ that is not invertible and a left-right $(A,H)$-Hopf module whose corresponding homomorphism $M otimes H to (A otimes H) $ is not an isomorphism.