I myself think I understand quite well also in more general settings the quotient W(0)\mapsto 0, which is described as quantum group representations. But I do not understand the effect by which the W(0) enters, and this is what I ask about.

Obviousely, thank you alot for the help: I am interested in the category with conjectured (!) tensor product in arXiv:0905.0916. On p.17-19 they describe the effect I asked about for the Virasoro algebra, p.38 they describe the same for the so-called logarithmic models, both by explicit lists of tensor products. There, the unit object is called V, the subobject J is called V(2), the quotient Q is called V(0), the extension the-other-ways-around is called V*. Respecitvely W,W(0),W(2),W*.

Thank you very much! The Nakayama-Argument with the splitting idempotent was unknown to me. I understand now, why this is not possible...But this leaves me puzzled: I know this category exists, yet it appearently cannot be realized as modules over a ring....? (I'll edit accordingly)

Though the more general approaches above were also very instructive to me...in particular abx comment on SL2(C) which I embarrassingly haven't seen. There of course I find all the groups I look for realiezed within different number fields.

Of course you are right, thank you. I have edited the question accordingly.