QUASI-QUANTUM GROUP COVARIANT Q-OSCILLATORS

If q is a pth root of unity there exists a quasi-co-associative truncated quantum group algebra U(q)T(sl2) whose indecomposable representations are the physical representations of U(q)(sl2), whose co-product yields the truncated tensor product of physical representations of U(q)(sl2), and whose R-matrix satisfies quasi Yang-Baxter equations. These truncated quantum group algebras are examples of weak quasi quantum group algebras [2]. For primitive pth roots q, q = e2pii/p, we consider a two-dimensional q-oscillator which admits U(q)T(sl2) as a symmetry algebra. Its wave function lie in a space F(q)T of ''functions on the truncated quantum plane'', i.e. of polynomials in noncommuting complex coordinate functions z(a), on which multiplication operators Z(a) and the elements of U(q)T(sl2) can act. This illustrates the concept of quasi quantum planes [1]. Due to the truncation, the Hilbert space of states is finite dimensional. The subspaces F(T(n)) of monomials in z(a) of nth degree vanish for n greater-than-or-equal-to p - 1, and F(T(n)) carries the (2J + 1)-dimensional irreducible representation of U(q)T(sl2) if n = 2J, J = 0, 1/2,..., 1/2(p - 2). Partial derivatives partial derivative a are introduced. We find a *-operation on the algebra of multiplication operators Z(i) and derivatives partial derivative b such that the adjoints Z(a)* act as differentiation on the truncated quantum plane. Multiplication operators Z(a) (''creation operators'') and their adjoints (''annihilation operators'') obey q-1/2-commutation relations. The *-operation is used to determine a positive definite scalar product on the truncated quantum plane F(q)T. Some natural candidates of hamiltonians for the q-oscillators are determined.