93-322 Flato M., Sternheimer D.

Star Products, Quantum Groups, Cyclic Cohomology and Pseudodifferential Calculus. (63K, plain TeX) Dec 4, 93

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Abstract. We start with a short historical overview of the developments of deformation (star) quantization on symplectic manifolds and of its relations with quantum groups. Then we briefly review the main points in the deformation-quantization approach, including the question of covariance (and related star-representations) and describe its relevance for a cohomological interpretation of renormalization in quantum field theory. We concentrate on the newly introduced notion of closed star product, for which a trace can be defined (by integration over the manifold) and is classified by cyclic (instead of Hochschild) cohomology ; this allows to define a character (the cohomology class of cocycle in the cyclic cohomology bicomplex). In particular we show that the star product of symbols of pseudodifferential operators on a compact Riemannian manifold is closed and that its character coincides with that given the trace, thus is given by the Todd class, while in general not satisfying the integrality condition. In the last section we discuss the relations between star products and quantum groups, showing in particular that "quantized universal enveloping algebras" (QUEAs) can be realized, essentially in a unique way (using a strong star-invariance condition) as star product algebras as star product algebras with a different quantization parameter. Finally we show (in the sl(2) case) that these QUEAs are dense in a model Frechet-Hopf algebra, stable under bialgebra deformations, containing all of them (for different parameter values) and that they have the same product and equivalent coproducts with the original algebra.

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