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Lately I've worked on an example that for a group $G$ the $G$-graded vector spaces $G$-vect and $G$-rep, the category of representations on $G$, are Morita equivalent.

I did that by taking the finite dimensional vector spaces vect$_k$ ($k$ a field) as a right module category over $G$-vect. Then I showed that $\mathrm{Fun}_{G-\mathrm{vect}}(\mathrm{vect}, \mathrm{vect})$ is equivalent to $G$-rep and by that those two are Morita equivalent.

It holds that $$k[G]-\mathrm{mod} \cong G-\mathrm{rep}$$ and $$k[G]-\mathrm{comod} \cong G-\mathrm{vect}.$$

One can now ask if a similar Morita equivalence holds for an arbitrary finite dimensional Hopf algebra $H$. It is known that $H-\mathrm{comod} \cong H^*-\mathrm{mod}$. So the question basically is:

Are $H-\mathrm{mod}$ and $H^*-\mathrm{mod}$ Morita equivalent?

As modules over a Hopf algebra are defined as vector spaces with an action (in my case at least) one can again take the vector spaces $\mathrm{vect}$ as the module category. But afterwards in my original proof I used especially representation-properties which I now don't have anymore.

Has someone worked with this before? I'm grateful for anything - tips on how to handle this, literature-tips or maybe even concrete tips on the proof (but I think maybe my question is too wide for this).

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  • $\begingroup$ Aren’t the categories of $G$-graded vector spaces and $G$-representations not morita equivalent, at least of $G$ finite and nonabelian? For example, the symmetric group $S_3$ has three simple representations, but six elements (and therefore six simple objects in the category of $S_3$-graded vector spaces)? $\endgroup$ – Joppy Jun 6 at 12:42
  • $\begingroup$ They are, as a reference take f.eks. in Etingofs Tensor categories Example 7.12.19. I'm not sure about your counter example but are you comparing the right things? for Morita equivalence the category of $S_3$-representations has to be isomorphic to the functor category $\mathrm{Fun}_{S_3-\mathrm{vect}}(\mathrm{vect}, \mathrm{vect})$. $\endgroup$ – P. Schulze Jun 6 at 15:46
  • $\begingroup$ Etingof does btw also say something about my current question, in Example 7.12.26, but I don't get what is happening there. $\endgroup$ – P. Schulze Jun 6 at 15:46
  • $\begingroup$ Are you really sure of what you are asking? I know that if $H$ is a finite dimensional Hopf algebra then you have an equivalence of categories $Comod^H\cong {_{H^*}Mod}$. Otherwise, in general, you only have that $Comod^H\cong \mathcal{Rat}\left(_{H^*}Mod\right)$ and that $comod^{H^\circ}\cong {_Hmod}$, where the capital letter denote the whole categories, while the small letter the subcategory of finite-dimensional ones. In particular, they hold for $G$ a finite group.. $\endgroup$ – Ender Wiggins Jun 8 at 9:22
  • $\begingroup$ yeah you are right, I still have the condition finite dimensional, I edited that $\endgroup$ – P. Schulze Jun 11 at 5:06

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