1,721,035 research outputs found
Leibniz Rules and Reality Conditions
No full textAbstract: An analysis is made of reality conditions within the context of noncommutative geometry. We show that if a covariant derivative satisfies a given left Leibniz rule then a right Leibniz rule is equivalent to the reality condition. We show also that the matrix which determines the reality condition must satisfy the Yang-Baxter condition if the extension of the covariant derivative to tensor products is to satisfy the reality condition. This is equivalent to the braid condition for the matrix which determines the right Leibniz rule
The geometry of the quantum Euclidean space
No full textAbstract: A detailed study is made of the noncommutative geometry of , the quantum space covariant under the quantum group . For each of its two -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 and as the limit Riemannian manifold
The hidden geometry of the quantum Euclidean space
No full textAbstract: We briefly describe how to introduce the basic notions of noncommutative differential geometry on the 3-dim quantum space covariant under the quantum group of rotations
Geometrical Issues for the 3-dim Quantum Euclidean Space
No full textAbstract: We briefly describe our application of a version of noncommutative differential geometry to the 3-dim quantum space covariant under the quantum group of rotations and sketch how this might be used to determine the correct physical interpretation of the geometrical observables
Leibniz Rules and Reality Conditions
No full textAbstract: An analysis is made of reality conditions within the context of noncommutative geometry. We show that if a covariant derivative satisfies a given left Leibniz rule then a right Leibniz rule is equivalent to the reality condition. We show also that the matrix which determines the reality condition must satisfy the Yang-Baxter condition if the extension of the covariant derivative to tensor products is to satisfy the reality condition. This is equivalent to the braid condition for the matrix which determines the right Leibniz rule
The hidden geometry of the quantum Euclidean space
No full textAbstract: We briefly describe how to introduce the basic notions of noncommutative differential geometry on the 3-dim quantum space covariant under the quantum group of rotations
Geometrical Issues for the 3-dim Quantum Euclidean Space
No full textAbstract: We briefly describe our application of a version of noncommutative differential geometry to the 3-dim quantum space covariant under the quantum group of rotations and sketch how this might be used to determine the correct physical interpretation of the geometrical observables
Frame formalism for the N-dimensional quantum Euclidean spaces
No full textAbstract: We sketch our recent application of a non-commutative version of the Cartan `moving-frame' formalism to the quantum Euclidean space , the space which is covariant under the action of the quantum group . For each of the two covariant differential calculi over based on the -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 . 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 which we use to project this frame in
Metrics on the Real Quantum Plane
No full textAbstract: Using the frame formalism we determine some possible metrics and metric-compatible connections on the noncommutative differential geometry of the real quantum plane. By definition a metric maps the tensor product of two 1-forms into a `function' on the quantum plane. It is symmetric in a modified sense, namely in the definition of symmetry one has to replace the permutator map with a deformed map \sigma fulfilling some suitable conditions. Correspondingly, also the definition of the hermitean conjugate of the tensor product of two 1-forms is modified (but reduces to the standard one if \sigma coincides with the permutator). The metric is real with respect to such modified *-structure
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