  We study realizations of polynomial deformations of the sl(2,R)- Lie algebra
in terms of differential operators strongly related to bosonic operators. We
also distinguish their finite- and infinite-dimensional representations. The
linear, quadratic and cubic cases are explicitly visited but the method works
for arbitrary degrees in the polynomial functions. Multi-boson Hamiltonians are
studied in the context of these ``nonlinear'' Lie algebras and some examples
dealing with quantum optics are pointed out.
