Quasi quantum group covariant q-oscillators

Abstract

If q is a pth rooty of unit there exists a quasi-co-associative truncated quantum group algebra U q T(sl 2) whose indecomposable representations ar the physical representations of U q (sl 2), whose co-product yields the truncated tensor product of physical representations of U q (sl 2), 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 = e 2 πi/ p , we consider a two-dimensional q-oscillator which admits U q T(sl 2) 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(sl 2) 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 R(n) of monomials in z a of nth degree vanish for n⩾ p − 1, and F T(n) carries the (2 J+1)-dimensional irreducible representation of U q T ( sl 2) 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, Z i and derivatives ∂ 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.