Let $A, B$ be finitely generated (noncommutative) algebras over a field $k$ (say, algebraically closed). Can we get all irreducible representations of $A \otimes_k B$ from tensoring representations of $A$ with representations of $B$? I'm especially interested in the case where $A, B$ are the enveloping algebras of finite-dimensional Lie algebras.
This is true for finite-dimensional representations, but false for infinite-dimensional ones. See pg. 31-32 of these lecture notes.