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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?

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    $\begingroup$ 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. $\endgroup$ – Sebastien Palcoux Aug 23 '19 at 17:30
  • $\begingroup$ @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? $\endgroup$ – Soap Aug 24 '19 at 12:21
  • $\begingroup$ @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? $\endgroup$ – Soap Aug 24 '19 at 12:24
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    $\begingroup$ You will find a scan of Loke's thesis in this link. About mine, see Recall 12.39 and Theorem 12.46. $\endgroup$ – Sebastien Palcoux Aug 24 '19 at 12:55
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    $\begingroup$ and yes, this answer should also help. $\endgroup$ – Sebastien Palcoux Aug 24 '19 at 13:10

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