# Monoidal category of irreducible highest weight modules of the Virasoro algebra

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?

• I am not sure to understand what do you want, but usually the irreducible representations of some $\mathfrak{g}$ corresponds to the simple objects 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. – Sebastien Palcoux Aug 23 '19 at 17:30
• @SebastienPalcoux This seems helpful, thank you. The link you provided does not contain the thesis itself, and I did not find it online. Is there a place where I can find the document? – Soap Aug 24 '19 at 12:21
• @SebastienPalcoux I did find your thesis, though. But on a first skim I only found comments about the category for the case of loop algebras, not the Virasoro algebra. Maybe I missed something? – Soap Aug 24 '19 at 12:24
• You will find a scan of Loke's thesis in this link. About mine, see Recall 12.39 and Theorem 12.46. – Sebastien Palcoux Aug 24 '19 at 12:55
• and yes, this answer should also help. – Sebastien Palcoux Aug 24 '19 at 13:10