Algebraization and exact solution of a master Hamiltonian of bosons and fermions

We introduce novel polynomial deformations of the 3-dimensional $A_1$ algebras, which give rise to an algebraization of a very general Hamiltonian of interest in atomic, molecular, nuclear and optical physics. We construct the unitary representations and the corresponding differential operator realizations of the polynomial algebras. This enables us to transform the Hamiltonian into a higher order differential operator which is quasi-exactly solvable. We solve the Hamiltonian differential equation by the functional Bethe ansatz, thus obtaing the exact solutions of the general Hamiltonian. This includes as special cases solutions of many interesting models such as the Bose-Einstein condensate models, the Lipkin-Meshkov-Glick model and the Tavis-Cummings model.