Quasi quantum group covariant q-oscillators

If q is a pth root of unity there exists a quasi-co-associative truncated quantum group algebra UqT(sl2) whose indecomposable representations are the phyiscal representations of Uq(sl2), whose co-product yields the truncated tensor product of physical representations of Uq(sl2), and whose R-matrix satisfies quasi Yang-Baxter equations. For primitive pth roots, q, q=e2πi/p, we consider a two-dimensional q-oscillator which admits UqT(sl2) as a symmetry algebra. Its wave function lie in a space FqT of 'functions on the truncated quantum plane', i.e. of polynomials in noncommuting complex coordinate functions za, on which multiplication operators Za and the elements of UqT(sl2) can act. Due to the truncation, the Hilbert space of states is finite dimensional. The subspaces FT(n) of monomials in za of nth degree vanish for n≥p-1, and FT(n) carries the (2J+1)-dimensional irreducible representation of UqT(sl2) if n=2J, J=0, 1/2, .., 1/2 (p-2). Partial derivatives ∂a are introduced. We find a *-operation on the algebra of multiplication operators Zi and derivatives ∂b such that the adjoints Za* act as differentiation on the truncated quantum plane. Multiplication operators Za ('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 FqT. Some natural candidates of hamiltonians for the q-oscillators are determined. (orig./HSI)