I'm trying to see if I can construct a monoidal category $\mathbf{C}$ whose objects are the irreducible unitary highest weight representations of the Virasoro algebra.

I am thinking on doing the following, which seems quite natural: start with the category $\mathbf{Rep}(Vir)$ and let $\mathbf{C}$ the full subcategory whose objects are the irreducible unitary highest weight representations.

But now if we take the tensor product to be the usual tensor product of representations of a Lie algebra, then the tensor product of two objects of $\mathbf{C}$ may be reducible in general, and thus the category is not monoidal. (This is my only issue, since the tensor product of arrows will be trivially in $\mathbf{C}$ because this subcategory is full, and also $\mathbf{1}$ (the unit of the monoidal category $\mathbf{Rep}(Vir)$) is in $\mathbf C$).

Is there a way to fix this problem?

simpleobjects of $Rep(\mathfrak{g})$. So the tensor product of two simple objects is still an object of $Rep(\mathfrak{g})$. Now the monoidal category structure for such representations of $Vir$ was studied (in the Connes fusion framework) in the thesis of Terence Loke (and in mine for $Vir_{1/2}$). They are tensor product of $Rep$ of some loop algebras. $\endgroup$yes, this answer should also help. $\endgroup$4more comments