Irreducible representations of a tensor product

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.

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Could this also be tagged tensor-products? –  Rudy the Reindeer Apr 12 '12 at 8:35
@Matt: Sure. (and extra characters) –  Akhil Mathew Apr 13 '12 at 1:50