Let $\hat g$ be an untwisted affine Kac Moody algebra. I want to know if the tensor product of two irreducible highest weight modules of $\hat g$ is itself an irreducible highest weight module (IHWM) of $\hat g$.
From what I gathered from [1] and [2], every IHWM of $\hat g$ can be written in the form $L(\Lambda) := M(\Lambda)/J(\Lambda)$, where $J(\Lambda)$ is the unique maximal submodule of the Verma module $M(\Lambda)$.
Thus, my question can be rephrased as follows: is it true that
$$
\left(M(\Lambda_1)/J(\Lambda_1)\right)\otimes \left(M(\Lambda_2)/J(\Lambda_2)\right) = M/J
$$
for some Verma module $M$ with maximal submodule $J$?
I am a novice in terms of properties of the tensor product and quotients of modules (in particular for modules over Lie algebras as in this case), and I do not know how these two operations interact. Are there any easy identities which allow me to arrive at the result?
[1] Roger Carter and Roger William Carter. Lie algebras of finite and affine
type, volume 96. Cambridge University Press, 2005.
[2] Victor G. Kac. Infinite-Dimensional Lie Algebras. Cambridge University
Press, 3 edition, 1990.