Projective module description of embedded noncommutative spaces

TL;DR

This paper provides an algebraic framework for embedded noncommutative spaces over the Moyal algebra, explicitly constructing projective modules for tangent bundles and deriving geometric structures like metrics and connections.

Contribution

It introduces an algebraic approach to noncommutative geometry, explicitly constructing tangent modules and deriving geometric quantities from algebraic data.

Findings

Constructed projective modules for tangent bundles.

Derived metric, Levi-Civita connection, and curvatures algebraically.

Discovered a bar involution with implications for noncommutative geometry.

Abstract

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.