Projective module description of embedded noncommutative spaces

Details Projective module description of embedded noncommutative spaces Projective module description of embedded noncommutative spaces

2008-10-14

arxiv.org/abs/0810.2357v2

Physics

Mathematical Physics

25 pages, final version, to appear in Reviews in Mathematical Physics

Scientific paper

An algebraic formulation is given for the embedded noncommutative spaces over the Moyal algebra developed in a geometric framework in \cite{CTZZ}. 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 \cite{CTZZ}. 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.

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