Quantum integrable models of interacting bosons and classical r-matrices with spectral parameters

Using the technique of classical $r$-matrices with spectral parameters we construct a general form of quantum Lax operators of interacting boson systems corresponding to an arbitrary simple (or reductive) Lie algebra. We prove quantum integrability of these models in the physically important case of $\mathfrak{g}=gl\left(n\right)$ and “diagonal” in the root basis classical $r$-matrices. We consider in detail two classes of non-skew-symmetric classical $r$-matrices with spectral parameters and obtain the corresponding quantum Lax operators and quantum integrable many-boson hamiltonians that generalize Bose–Hubbard dimer hamiltonians.