Variations on deformation quantization

TL;DR

This paper explores key aspects of deformation quantization, including the uniqueness of universal star products, cohomology classes for differential star products, and the construction of convergent star products on Hermitian symmetric spaces, providing a historical overview.

Contribution

It discusses the uniqueness, classification, and construction of star products in deformation quantization, highlighting new insights and methods in these areas.

Findings

Proof of the uniqueness of universal star products on duals of Lie algebras.

Analysis of Deligne's cohomology classes for symplectic manifolds.

Construction of convergent star products on Hermitian symmetric spaces.

Abstract

I have chosen, in this presentation of Deformation Quantization, to focus on 3 points: the uniqueness --up to equivalence-- of a universal star product (universal in the sense of Kontsevich) on the dual of a Lie algebra, the cohomology classes introduced by Deligne for equivalence classes of differential star products on a symplectic manifold and the construction of some convergent star products on Hermitian symmetric spaces. Those subjects will appear in a promenade through the history of existence and equivalence in deformation quantization.