 93322 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 deformationquantization approach, including the
question of covariance (and related starrepresentations)
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 starinvariance 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 FrechetHopf 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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