PROJECTIVE MODULE DESCRIPTION OF EMBEDDED NONCOMMUTATIVE SPACES

An algebraic formulation is given for the embedded noncommutative spaces over the Moyal algebra developed in a geometric framework in [8]. 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 [8]. 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.

SUPERSYMMETRIC EXTENSION OF MOYAL ALGEBRA AND ITS APPLICATION TO THE MATRIX MODEL

We construct operator representation of Moyal algebra in the presence of fermionic fields. The result is used to describe the matrix model in Moyal formalism that treat gauge degrees of freedom and outer degrees of freedom equally.

A q-DEFORMED VERSION OF THE HEAVENLY EQUATIONS

Using a q-deformed Moyal algebra associated with the group of area-preserving diffeomorphisms of the two-dimensional torus T2, sdiffq(T2), a q-deformed version of the heavenly equations is given. In addition, the two-dimensional chiral version of self-dual gravity in this q-deformed context is briefly discussed.