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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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