q-ANALOGUE OF BOSON COMMUTATOR AND THE QUANTUM GROUPS SUq(2) AND SUq(1, 1)

We propose a defining set of commutation relations to a q-analogue of boson operator; [Formula: see text], [Formula: see text] and [N, aq]=−aq, which contracts to the Heisenberg algebra of boson operators in the limit of q=1. Here, N is the number operator, [N]q being its q-analogue operator. By making use of this set, we construct a new realization of the “noncompact” quantum group SUq(1, 1) in addition to that of the SUq(2) recently proposed by Biedenharn. The explicit form of the number operator is given in terms of aq and [Formula: see text] and its positive definiteness is proved. A uniqueness of our commutators is also discussed. It is shown that the quantum group SUq(2) appears as a true symmetry group of a q-analogue of the two-dimensional harmonic oscillator and the SUq(1, 1) as its dynamical group.