## Introduction to noncommutative differential geometry.(English)Zbl 0989.53059

In the framework of star product, this article gives a survey of deformation quantization which both gives motivation and provides important examples to the study of noncommutative differential geometry. The starting point is a review of the abstract theory of Moyal product.
In the following, we denote by a triple $$( A,A',\ast _{0})$$ a complete linear space $$A$$ which is a (topological) $$A'$$-bimodule with the algebra $$A'$$ embedded in $$A$$ as a dense $$A'$$-bimodule and with $$\ast _{0}$$ as the multiplication operation involved. For any commuting derivations $$D,D':A\to A$$ of the $$A'$$-bimodule $$A$$ which keep $$A'$$ invariant and are quasinilpotent (or pointwise nilpotent), we can deform the bimodule structure $$( A,A',\ast _{0})$$ to another one $$( A,A',\ast _{1})$$ by defining the Moyal product $$f\ast _{1}g:=f\exp ( \hbar \overleftarrow{D} \ast _{0}\overrightarrow{D'}) g$$ where $$\hbar >0$$ and $$f( \overleftarrow{D}\ast _{0}\overrightarrow{D'}) ^{k}g:=( D^{k}f) \ast _{0}( D^{\prime k}g)$$. This kind of process can be repeated and in particular, we can get the deformed bimodule $$( A,A',\ast)$$, where $$\ast$$ is the star product (determined by $$( D,D')$$ and $$\ast _{0}$$) defined by $$f\ast g:=f\exp ( \hbar \overleftarrow{D}\ast _{0} \overrightarrow{D'}-\hbar \overleftarrow{D'}\ast _{0} \overrightarrow{D}) g$$.
Along this general approach, the well-known Weyl algebra, in both Cartesian and polar coordinate systems, and the algebras of noncommutative 2-sphere and quantum $$SU_{q}( 2)$$ are presented and analyzed. Furthermore deformation quantization is studied in the context of $$\mu$$-related algebras, and in particular, the contact algebra $$C^{\infty }( \mathbb{S}^{3})$$ is deformation quantized in this way.

### MSC:

 53D55 Deformation quantization, star products 58B32 Geometry of quantum groups 81S10 Geometry and quantization, symplectic methods 81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
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