Also in fact I don't know much about representation theory as I'm more a probabilist, could you recommend a reference about all these tools if it's not annoying ? Thank you so much, it is of big help!

Thank you so much for your long answer! If I summarize: - I should think about the monodromy operator and the R-matrix as encoding the generators of the Hopf-algebra called the Yangian - The local Lax operator is the image of the monodromy matrix via the finite dimensional representation of the Lie Alegbra Sl_2. Is it correct ?