Metrics on the Real Quantum Plane

Abstract

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 σfulfilling 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 σcoincides with the permutator). The metric is real with respect to such modified *-structure.

21 pages, no figures. Revised version to appear in JMP

References in corpus (7)

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• Linear connections on the quantum plane
• Bicovariant Calculus on Twisted ISO(N), Quantum Poincare' Group and Quantum Minkowski Space
• Solutions of Klein--Gordon and Dirac equations on quantum Minkowski spaces
• The Geometry of a $q$-Deformed Phase Space