On the Fedosov Deformation Quantization beyond the Regular Poisson Manifolds

TL;DR

This paper introduces an iterative algebraic method for deformation quantization of irregular Poisson brackets linked to the classical Yang-Baxter equation, providing a universal formula applicable to triangular Lie bialgebras.

Contribution

It presents a new algebraic approach to deformation quantization beyond regular Poisson manifolds, including a universal deformation formula and classification results.

Findings

Provides an explicit quantization formula for quasi-homogeneous Poisson brackets on the two-plane.

Establishes a universal deformation formula for any triangular Lie bialgebra.

Offers a simple proof for the classification of inequivalent universal deformation formulas.

Abstract

A simple iterative procedure is suggested for the deformation quantization of (irregular) Poisson brackets associated to the classical Yang-Baxter equation. The construction is shown to admit a pure algebraic reformulation giving the Universal Deformation Formula (UDF) for any triangular Lie bialgebra. A simple proof of classification theorem for inequivalent UDF's is given. As an example the explicit quantization formula is presented for the quasi-homogeneous Poisson brackets on two-plane.