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Star products, quantum groups, cyclic cohomology and pseudodifferential calculus. (English) Zbl 0827.17019
Sally, Paul J. jun. (ed.) et al., Mathematical aspects of conformal and topological field theories and quantum groups. AMS-IMS-SIAM summer research conference, June 13-19, 1992, Mount Holyoke College, South Hadley, MA, USA. Providence, RI: American Mathematical Society. Contemp. Math. 175, 53-72 (1994).
This paper is a survey, without proofs but with extensive references, of the topics listed in the title. The paper begins with a historical review and then reviews the main points of star-product theory. It then discusses the notion of closed star-products, for which a trace can be defined by integration over phase space. The authors show their relation to cyclic cohomology and apply it to pseudodifferential calculus on a compact Riemannian manifold. They conclude with a discussion of the relations between star-products and quantum groups, showing that quantized universal enveloping algebras are realized as star-product algebras in an essentially unique way.
Star-products are defined as follows: Let $$W$$ be a finite dimensional manifold and $$N$$ the algebra of $$C^\infty$$ functions on $$W$$. Let $$\Lambda$$ be a contravariant skew-symmetric 2-tensor on $$W$$ such that the Poisson bracket $$P(u,v)= i(\Lambda) (du\wedge dv)$$, $$u,v\in \mathbb{N}$$, satisfies the Jacobi identity. Then a star-product is an associative deformation of $$N$$ with a complex parameter $$\nu$$: $u *v= \sum_{r=0}^\infty \nu^r C_r (u,v); \qquad u,v\in \mathbb{N},$ where $$C_0 (u,v)= uv$$; $$C_1 (u,v)- C_1 (v,u)= 2P (u,v)$$ and all the $$C_r$$ are bidifferential operators.
For the entire collection see [Zbl 0801.00049].
Reviewer: A.R.Magid (Norman)

##### MSC:
 17B37 Quantum groups (quantized enveloping algebras) and related deformations 81R50 Quantum groups and related algebraic methods applied to problems in quantum theory 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 58J40 Pseudodifferential and Fourier integral operators on manifolds 46L85 Noncommutative topology 46L87 Noncommutative differential geometry