An explicit two step quantization of Poisson structures and Lie bialgebras

TL;DR

This paper introduces a novel two-step explicit method for quantizing Poisson structures and Lie bialgebras, providing universal formulas and configuration space models, with the first step requiring an associator choice.

Contribution

It presents a new explicit two-step approach to deformation quantization of Lie bialgebras and Poisson structures, including universal formulas and configuration space models.

Findings

Explicit transcendental formulas for the two-step quantization process.

Configuration space models for biassociahedron and bipermutohedron.

Universal quantization formulas using smooth differential forms.

Abstract

We develop a new approach to deformation quantizations of Lie bialgebras and Poisson structures which goes in two steps. In the first step one associates to any Poisson (resp. Lie bialgebra) structure a so called quantizable Poisson (resp. Lie bialgebra) structure. We show explicit transcendental formulae for this correspondence. In the second step one deformation quantizes a quantizable Poisson (resp. Lie bialgebra) structure. We show again explicit transcendental formulae for this second step correspondence (as a byproduct we obtain configuration space models for biassociahedron and bipermutohedron). In the Poisson case the first step is the most non-trivial one and requires a choice of an associator while the second step quantization is essentially unique, it is independent of a choice of an associator and can be done by a trivial induction. We conjecture that similar statements hold true in the case of Lie bialgebras. The main new result is a surprisingly simple explicit universal formula (which uses only smooth differential forms) for universal quantizations of finite-dimensional Lie bialgebras.