Let $A$ be a Hopf algebra over the complex numbers. Denote by $\mathcal{M}$ the dg-category of dg-$A$-modules. The Hochschild homology of $\mathcal{M}$ is not going to be the Hochschild homology of $A$, but can I compute the Hochschild homology of $A$ as the Hochschild homology of a subcategory of $\mathcal{M}$, preferrably some subcategory of objects which are dualizable?
I know that I can take perfect modules, but this subcategory, for instance, might not contain the tensor unit of $\mathcal{M}$, which might be a problem.
In summary, my question is what are the dg-categories built from $A$ and its modules whose Hochschild homology gives me the Hochschild homology of $A$?
