6533b820fe1ef96bd1279927

RESEARCH PRODUCT

Quantum Groups, Star Products and Cyclic Cohomology

Daniel SternheimerMoshé Flato

subject

AlgebraStar productPhase spaceCyclic homologyUniquenessRiemannian manifoldQuantumAtiyah–Singer index theoremMathematicsBialgebra

description

After some historical remarks, we start with a rapid overview of the star-product theory (deformation of algebras of functions on phase space) and its applications to deformation-quantization. We then concentrate on Poisson-Lie groups and their “quantization”, give a star-product realization of quantum groups and discuss uniqueness and the rigidity as bialgebra of a universal model for the quantum SL(2) groups. In the last part we develop the notion of closed star-product (for which a trace can be defined on the algebra), show that it is classified by cyclic cohomology, permits to define a character and that there always exists one; finally we show that the pseudodifferential calculus on a compact Riemannian manifold defines a closed star product whose character contains the Todd genus of the manifold.

yearjournalcountryeditionlanguage
1993-01-01
https://doi.org/10.1007/978-94-017-2823-2_17