Realization of the Three-dimensional Quantum Euclidean Space by Differential Operators

The three-dimensional quantum Euclidean space is an example of a non-commutative space that is obtained from Euclidean space by $q$-deformation. Simultaneously, angular momentum is deformed to $so_q(3)$, it acts on the $q$-Euclidean space that becomes a $so_q(3)$-module algebra this way. In this paper it is shown, that this algebra can be realized by differential operators acting on $C^{\infty}$ functions on $\mathbb{R}^3$. On a factorspace of $C^{\infty}(\mathbb{R}^3)$ a scalar product can be defined that leads to a Hilbert space, such that the action of the differential operators is defined on a dense set in this Hilbert space and algebraically self-adjoint becomes self-adjoint for the linear operator in the Hilbert space. The self-adjoint coordinates have discrete eigenvalues, the spectrum can be considered as a $q$-lattice.Comment: 13 pages, late

The Geometry of a qq-Deformed Phase Space

The geometry of the $q$-deformed line is studied. A real differential calculus is introduced and the associated algebra of forms represented on a Hilbert space. It is found that there is a natural metric with an associated linear connection which is of zero curvature. The metric, which is formally defined in terms of differential forms, is in this simple case identifiable as an observable.Comment: latex file, 26 pp, a typing error correcte

The Geometry of the Quantum Euclidean Space

A detailed study is made of the noncommutative geometry of $R^3_q$, the quantum space covariant under the quantum group $SO_q(3)$. For each of its two $SO_q(3)$-covariant differential calculi we find its metric, the corresponding frame and two torsion-free covariant derivatives that are metric compatible up to a conformal factor and which yield both a vanishing linear curvature. A discussion is given of various ways of imposing reality conditions. The delicate issue of the commutative limit is discussed at the formal algebraic level. Two rather different ways of taking the limit are suggested, yielding respectively $S^2\times R$ and $R^3$ as the limit Riemannian manifold.Comment: 29 pages, latex fil

Frame formalism for the N-dimensional quantum Euclidean spaces

We sketch our recent application of a non-commutative version of the Cartan `moving-frame' formalism to the quantum Euclidean space $R^N_q$, the space which is covariant under the action of the quantum group $SO_q(N)$. For each of the two covariant differential calculi over $R^N_q$ based on the $R$-matrix formalism, we summarize our construction of a frame, the dual inner derivations, a metric and two torsion-free almost metric compatible covariant derivatives with a vanishing curvature. To obtain these results we have developed a technique which fully exploits the quantum group covariance of $R^N_q$. We first find a frame in the larger algebra \Omega^*(R^N_q) \cocross \uqs. Then we define homomorphisms from R^N_q \cocross U_q^{\pm}{so(N)} to $R^N_q$ which we use to project this frame in $\Omega^*(R^N_q)$.Comment: Latex file, 11 pages. Talks given at the Euroconference ``Non-commutative Geometry and Hopf Algebras in Field Theory and Particle Physics'', Villa Gualino (Torino), Sept. 199