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].

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 |