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 $\_B\mathcal{M}$?



Category of representations of a tensor product algebra
Yes, it will be exactly Deligne's tensor product of abelian categories. See <https://ncatlab.org/nlab/show/Deligne+tensor+product+of+abelian+categories>


