Polynomial deformations of the Lie algebras $sl(2)$ in problems of quantum optics

It is shown that specific (polynomial) deformations of Lie algebras arise naturally as dynamical symmetry algebras $g^{ds}$ of second-quantized models with nonquadratic Hamiltonians $H$ invariant with respect to certain groups $G^{\text {inv}}(H)$. Such deformations $sl_ d(2)$ of the Lie algebras $sl(2)$ are found in a number of models of quantum optics (multiphoton processes, generalized Dicke model, and frequency conversion), and ways to apply thes $sl(2)$ formalism to the solution of physics problems are indicated.