@leibnewtz It depends where you read it. That's why I said "that version", which says that for finite dimensional irreducible modules one has that elements of the center are represented as multiples of the identity. You can see this as a corollary of the Schur's lemma you are thinking about. I am wondering if this corollary somehow extends to infinite-dimensional modules.

@StevenLandsburg I added it explicitly.

@paulgarrett So in this case the assumption would be that the derivation extension does not affect the main properties, it seems. That seems to be about right: in Kerf's book (a mathematical physicist from what I gather) he says that the derivation extension is a "technicality".