Are there interesting semisimple algebras in non-semisimple categories?

Question

Are there any interesting examples of semisimple algebras in nonsemisimple categories which don't "come from" a semisimple algebra in a semisimple category? That is, if you want to study semisimple algebra objects can you assume wlog that the underlying category is semisimple?

Here's one way of trying to make this question precise. Suppose that C is a finite tensor category, and A is an algebra object in C. Further suppose that A-mod (the category of left A-module objects in C) is a semisimple category. Does there always exist: a semisimple tensor category C', a monoidal functor F: C'->C, and an algebra object A' in C' such that F(A') = A? (Perhaps I should also require some relationship between A-mod and A'-mod, at the very least A'-mod should be semisimple.)

I would also accept an answer explaining why "A-mod is semisimple" doesn't correctly capture the notion of a semisimple algebra if C is non-semisimple.

Answer 1

If $H$ is a finite dimensional Hopf algebra and $\mathcal C=\mathcal M^H$ is the category of corepresentation then $H$ is an algebra in $\mathcal C$ and $\mathcal{C}_H=\mathcal M_H^H= Vec$, where the last equivalence follows by the fundamental theorem of Hopf modules. Then if $H$ is not semisimple (for example the Taft algebra) still $H$ is a semisimple (in fact simple) in the finite tensor category $\mathcal C$.

Note that $H$ as an object in $\mathcal C$ is a tensor generator, so using arguments of Tannakian reconstruction is possible to prove that you can not have a tensor functor $F:\mathcal C'\to \mathcal C$ and an algebra $A'$ in $\mathcal C'$ such that $F(A')=H$ with $\mathcal C'$ a semisimple tensor category.