Let $G$ be a group (note $G$ is not necessarily abelian), and let $A$ be a $G$-graded algebra over a field $k$. We know that a $k$-central $(A,A)$-bimodule corresponds to a (left) $A^e$-module via $a \cdot m \cdot b := (a \otimes b^{\text{op}}) \cdot m$. I would like for this type of correspondence to hold when we know that one of them is $G$-graded as well, i.e. a $G$-graded $k$-central $(A,A)$-bimodule $U$ is a $G$-graded (left) $A^e$-module.
Some thoughts:
- $A^e$ has a natural $G \times G$ grading given by $(A \otimes A^{\text{op}})_{(\sigma, \tau)} = A_{\sigma} \otimes A^{\text{op}}_{\tau}$, where $A^{\text{op}}_{\tau} = A_{\tau^{-1}}$.
- Any $G$-grading can be turned into a $G \times G$ grading by considering the diagonal map $\Delta: G \to G \times G$.
Any further ideas or insights are greatly appreciated.