Let us continue this discussion in chat.

@JoséFigueroa-O'Farrill Just one more thing: you say that in the Sugawara construction one starts with a representation of a Kac-Moody algebra, but in the physics literature they seem to start with a central extension of a loop algebra instead (so they do not use the derivation extension). So I would expect them to use the (integrable) reps of central extension of a loop algebra, not the (integrable) reps of the affine Kac Moody algebra, to classify the WZW models... but they use the latter.

@JoséFigueroa-O'Farrill That I know. I should have been more explicit: why integrable representations?

@JoséFigueroa-O'Farrill Why do we only use the integrable highest weight representations?

Thanks, I've been looking into that book. By the way, I just found out that this is discussed and proved in the book "Lie algebras with triangular decompositions" by Moody and Pianzola, page 33, in case someone reads this in the future.

Do you know of some book or text which expands a bit on this?

@SylvainRibault I honestly do not know what you mean. Models here are "coset models", a construction due to GKO of Virasoro representations. They have a certain central charge and highest weight. Do you know what a coset model is? (In conformal field theory)

@SebastienPalcoux My motivation comes from physics (long story), since there is another monoidal category which is conjectured to be related to this category of irreps. (I do not know about that Perron-Frobenius dimension). I am a physicist with not that much (maybe not enough) knowledge about tensor categories the like)

@SebastienPalcoux I now realized that in your first comment you referred something that is probably my biggest concern: the tensor product of two irreducible representations will not be irreducible in general, as I wrote in my question. So I want a "new tensor product" for which the product of irreps is still an irrep. Do you know if that Connes fusion product has this property?

@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?

@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?