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When, many years ago, I took a seminar class on this (based on, e.g. arxiv.org/abs/q-alg/9603025 ), I remember that many of the modules that arose naturally were indeed completely reducible, in a form very similar to the $q=0$ versions.
I think the real answer is that you should learn more about quantum mechanics and quantum field theory as general subjects. (The fact that you have never seen $\phi^{4}$ theory in $d=1$ indicates that you do not have much grounding in the physics.) If you are having trouble making the connections between apparently different formalisms used in different sources, that is probably because they assume a certain level of understanding of the physics that motivates the mathematics.
I included some thoughts on this in the first two sections of arXiv:1902.04643 . It doesn't get into the actual structure of the of the infinite hierarchy of conservation laws, but it discussed why, qualitatively, a soliton equation has to have countably many conserved quantities.
