Consider an Ab-category $\mathcal{D}$ and its pseudo-abelian envelope $\mathcal{C}$. If necessary, one can assume $\mathcal{D}$ to be $k$-linear over a field $k$ and have finite-dimensional hom-spaces. I am interested in conditions on $\mathcal{D}$ that will ensure that $\mathcal{C}$ is abelian. Are there any such nontrivial conditions? (For motivation, note for instance that the categories in, say, [this paper](http://arxiv.org/abs/math/0610552) are defined via such a pseudo-abelianization procedure.)


**Edit:** The answer below discusses the semisimple abelian case. What about the non-semisimple case?



When is the pseudo-abelian envelope abelian?
A sufficient (but not necessary) condition is that the endomorphism ring of every object be semi-simple. See [Jannsen](http://www.ams.org/mathscinet-getitem?mr=1150598).


