Artin-Wederburn says that if you have a semisimple algebra then it is a product of matrix algebras over division rings.


Suppose that $C$ is a fusion category over the complex numbers (if you want to assume pivotal or similar things, that's fine, but don't assume symmetric or braided). Suppose that $X$ is an algebra object in $C$. That is $X$ is an object in $C$ together with a multiplication map $X \otimes X \rightarrow X$ and a unit etc. We call $X$ semisimple if the category of $X$-module objects in $C$ is semisimple. Is there some good analogue of Artin-Wedderburn?



Is there a good version of Artin-Wedderburn for semisimple algebra objects?
I have a paper on that topic:


* *Finite, connected, semisimple, rigid tensor categories are linear*, Math. Res. Lett. **10** (2003), 411-421, doi:[10.4310/MRL.2003.v10.n4.a1](https://doi.org/10.4310/MRL.2003.v10.n4.a1), arXiv:[math/0209256](https://arxiv.org/abs/math/0209256).


