Polynomial deformations of the Lie algebrasl(2) in problems of quantum optics

It is shown that specific (polynomial) deformations of Lie algebras arise naturally as dynamical symmetry algebrasg^ds of second-quantized models with nonquadratic HamiltoniansH invariant with respect to certain groupsG^inv(H). Such deformationssl_d(2) of the Lie algebrasl(2) are found in a number of models of quantum optics (multiphoton processes, generalized Dicke model, and frequency conversion), and ways to apply thesl(2) formalism to the solution of physics problems are indicated.P. N. Lebedev Physics Institute, Russian Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 95, No. 1, pp. 3–19, April, 1993.