Polynomial Lie algebras in solving a class of integrable models of quantum optics: exact methods and quasiclassics

A wide class of integrable quantum-optical models with G_i-invariant Hamiltonians H is described in the form when H are linear functions in generators of the polynomial Lie algebras su_pd(2) and Hilbert spaces L(H) of quantum states are decomposed in direct sums of su_pd(2)-irreducible subspaces. This yields exact and approximate methods of solving physical problems as well as new (su_pd(2)-cluster) quasiclassics in original models.