v 1 1 9 A ug 1 99 3 q - Oscillators , q - Epsilon Tensor , q - Groups Metin Arik ,

Considering a multi-dimensional q-oscillator invariant under the (non quantum) group U(n), we construct a q-deformed Levi-Civita epsilon tensor from the inner product states. The invariance of this q-epsilon tensor is shown to yield the quantum group SLq(n) and establishes the relationship of the U(n) invariant q-oscillator to quantum groups and quantum group related oscillators. Furthermore the q-epsilon tensor provides the connection between SLq(n) and the volume element of the quantum hyper plane. With the discovery of the quantum groups [1], q-generalizations of the harmonic oscillator have recently been the center of attention [2]. There exist different formulations of these generalizations, the so called q -oscillators. Strict constraints on these generalizations come from the multidimensionality of the oscillator. One aspect that has been considered is invariance under the unitary quantum group [3]. A related aspect that yields quantum group invariance is the degeneracy [4] of the eigenstates of the generalized bilinear hamiltonian.