Consider the category $\mathcal O$ for a Lie algebra $\mathfrak g$. As I understand it, we can define such a category whenever $\mathfrak g$ has a triangular decomposition. So $\mathfrak g$ can be for example a semisimple Lie algebra, or an affine Kac Moody algebra, or the Virasoro algebra.
We can construct Verma modules as quotients of the universal enveloping algebra, and they are objects of $\mathcal O$ [1,2]. So are their tensor products and their quotients.
In particular, all the highest weight irreducible representations $L(\Lambda)$ are in $\mathcal O$, since they are quotients of Verma modules.
This means that we can construct a category of irreducible highest weight representations as a full subcategory of $\mathcal O$. Let's call this category $C$.
My Question:
How can I make this category $C$ a monoidal category? (What tensor product can I define?)
Notice that the usual tensor product does not work, since the tensor product of irreps is in general not an irrep.
[1] Kac, V. G. (1990). Infinite-Dimensional Lie Algebras. Cam-
bridge University Press, 3 edition.
[2] Kac, V. G., Raina, A. K., and Rozhkovskaya, N. (2013).
Bombay lectures on highest weight representations of infinite dimensional
Lie algebras, volume 29. World scientific.