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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?

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I have a paper on that topic:

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  • $\begingroup$ Hrm, I think I need to spend some more serious time with that paper. I think this is not the first time (and maybe not the second either) that I've asked a question and you've said the answer was in this paper. I even read it after one of those times. $\endgroup$ Oct 1 '09 at 19:43
  • $\begingroup$ Ok, so certainly this is some form of Artin-Wedderburn, but I don't think it's of the form I was looking for. Your result says that just as semisimple rings are always algebras over a field fusion rings are also always linear over some field. But what I wanted wasn't something that replaced semisimple rings with fusion categories but rather something that replaced semisimple rings in Vect with semisimple rings in some other category. $\endgroup$ Oct 7 '09 at 20:32

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