TITLE: Is there a good version of Artin-Wedderburn for semisimple algebra objects?
QUESTION [8 upvotes]: 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?

REPLY [4 votes]: 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, arXiv:math/0209256.