Given two semisimple unital algebras $A$ and $B$, defined over $\mathbb{R}$ or $\mathbb{C}$, denote their categories of representations by $_A\mathcal{M}$ and $_B\mathcal{M}$ respectively. Can one describe the category of representations of $A \otimes_{\mathbb{C}} B$ as some type of "tensor product" of the categories $_A\mathcal{M}$ and $_B\mathcal{M}$?
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asked Jan 4, 2020 at 17:24
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4$\begingroup$ There is a notion of Deligne tensor product of tensor categories, probably that's what you need. $\endgroup$– Victor PetrovCommented Jan 4, 2020 at 18:33
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2$\begingroup$ See also some discussion here: mathoverflow.net/questions/335810/… $\endgroup$– Victor PetrovCommented Jan 4, 2020 at 18:45
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1 Answer
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Yes, it will be exactly Deligne's tensor product of abelian categories. See https://ncatlab.org/nlab/show/Deligne+tensor+product+of+abelian+categories
answered Jan 6, 2020 at 12:06