Classical vs. Quantum Groups as Symmetries of Quantized Systems

Quantization of nonlinear systems has been one of the fundamental problems of quantum mechanics. The idea that classical integrable nonlinear systems should be solvable after quantization, has lead to the introduction of quantum groups as “generalized symmetries” of a physical system. Quantum groups [1] can he obtained by applying the idea of q-deformation either to the Lie algebra or to a matrix representation of the corresponding classical group. However, as we will show a q-deformation of the algebra of quantum observables does not necessarily imply a q-deformation of the symmetry group. To clarify this point, we discuss the quantum group invariant and the classical group invariant q-oscillators separately in the second section. In the third section, we consider a specific quadratically constrained model [2] and show that the classical group invariant q-oscillator can be derived using Dirac bracket (diracket) quantization [3] with some modifications.