Algebraic Noncommutative Geometry

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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 A⁰ of algebraic functions on an algebraic manifold M is obtained. This A⁰ is a subalgebra of C(M), which is dense if M is compact. The generators of A define an immersion of M into Rⁿ, 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 Rⁿ, T⋆ S², T², S² 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.