Symmetric spaces, non-formal star products and Drinfel'd twists

TL;DR

This paper reviews and extends methods for non-formal deformation quantization of symmetric spaces, introducing the Retract Method for constructing star products and deriving Drinfel'd twists for non-Abelian Lie group actions.

Contribution

It introduces the Retract Method for defining star products on symmetric spaces and derives non-formal Drinfel'd twists for non-Abelian Lie group actions, advancing noncommutative geometry techniques.

Findings

Retract Method effectively constructs star products on symmetric spaces.

Derived non-formal Drinfel'd twists for non-Abelian Lie groups.

Extended the framework of noncommutative deformation quantization.

Abstract

These notes refer to a minicourse I gave at the occasion of the conference meeting ``Applications of Noncommutative Geometry to Gauge Theories, Field Theories, and Quantum Space-Time'' to be held from 7 April to 11 April 2025 at the Centre International de Rencontres Math\'ematiques in Luminy. They consist in a review of a long standing work of mine and collaborators (see references therein) in the field of non-formal deformation quantization admitting a large group of symmetries. But they also contain new material and results. More precisely, in a first part, I present a method (called the Retract Method) to define quantizations/symbolic calculi and associated operator symbol composition formulae (non-formal deformations/star products) of symplectic symmetric spaces such as the hyperbolic plane (Kahler) or symmetric co-adjoint orbits of the Poincar\'e group (non-metric). In a second part, I explain how to derive non-formal Drinfel'd twists for actions of non-Abelian solvable Lie groups (non-Abelian Universal Deformation Formulae) on or Fr \'echet algebras from the non-formal noncommutative symmetric spaces defined in the first part.