Algebraic Noncommutative Geometry

Abstract

A noncommutative algebra $A$, called an algebraic noncommutative geometry, is defined, with a parameter $\epsilon$ in the centre. When $\epsilon$ is set to zero, the commutative algebra $A^0$ of algebraic functions on an algebraic manifold $M$ is obtained. This $A^0$ is a subalgebra of $C(M)$, which is dense if $M$ is compact. The generators of $A$ define an immersion of $M$ into $R^n$, and $M$ inherits a Poisson structure as the limit of the commutator. Thus $A$ is a quantisation of a Poisson manifold. If an ordering convention is prescribed for $A$ then a star product on $M$ is obtained. Homomorphism and isomorphisms between noncommutative geometries are defined, and the map from $A$ to the Heisenberg algebra is used both to give an analogue of a coordinate chart, and to give $A$ a quantum group structure. Examples of algebraic noncommutative geometries are given, which include $R^n$, $T^{\star }{S}^2$, $T^2$, $S^2$ and surfaces of rotation. A definition of a metric on $M$ which can be extended to noncommutative geometry is given and this is used in an application of noncommutative geometry to the numerical analysis of surfaces.

Latex 29 pages, no figures, submitted to Comm. Math. Physics