Algebraic Noncommutative Geometry

Abstract

A noncommutative algebra $A$, called an algebraic noncommutative geometry, is defined, with a parameter $ε$ in the centre. When $ε$ 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.