2008/10/14 by R. B. Zhang, Xiao Zhang, Zhang, R. B. +1 · 1 citation
Mathematics · Physics and Astronomy · #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph) #gr-qc #hep-th #math-ph #math.MP
paper · pdf · doi:10.48550/arxiv.0810.2357
25 pages, final version, to appear in Reviews in Mathematical Physics
arxiv created 2010/05/12 · arxiv updated 2010/05/13
An algebraic formulation is given for the embedded noncommutative spaces over the Moyal algebra developed in a geometric framework in \citeCTZZ. We explicitly construct the projective modules corresponding to the tangent bundles of the embedded noncommutative spaces, and recover from this algebraic formulation the metric, Levi-Civita connection and related curvatures, which were introduced geometrically in \citeCTZZ. Transformation rules for connections and curvatures under general coordinate changes are given. A bar involution on the Moyal algebra is discovered, and its consequences on the noncommutative differential geometry are described.