DEFORMED U(2) ALGEBRA AS THE SYMMETRY ALGEBRA OF THE PLANAR ANISOTROPIC QUANTUM HARMONIC-OSCILLATOR WITH RATIONAL RATIO OF FREQUENCIES

The symmetry algebra of the two-dimensional anisotropic quantum harmon ic oscillator with rational ratio of frequencies is identified as a de formation of the u(2) algebra. The finite dimensional representation m odules of this algebra are studied and the energy eigenvalues are dete rmined using algebraic methods of general applicability to quantum sup erintegrable systems. For labelling the degenerate states an ''angular momentum'' operator is introduced, the eigenvalues of which are roots of appropriate generalized Hermite polynomials. The cases with freque ncy ratios 1:n correspond to generalized parafermionic oscillators, wh ile in the special case with frequency ratio 2:1 the resulting algebra corresponds to the finite W algebra W-3((2)).