A Universal Construction of Universal Deformation Formulas, Drinfel'd Twists and their Positivity

TL;DR

This paper presents an explicit Fedosov-based construction of star products and Drinfel'd twists for Lie algebra modules, establishing a correspondence with cohomology and demonstrating positivity in Kähler cases.

Contribution

It provides a constructive method for universal deformation formulas and Drinfel'd twists, linking them to Lie algebra cohomology and positivity properties.

Findings

Explicit Fedosov construction of star products

One-to-one correspondence between twists and cohomology classes

Positive Drinfel'd twist for Kähler Lie algebras

Abstract

In this paper we provide an explicit construction of star products on U(g)-module algebras by using the Fedosov approach. This construction allows us to give a constructive proof to Drinfel'd theorem and to obtain a concrete formula for Drinfel'd twist. We prove that the equivalence classes of twists are in one-to-one correspondence with the second Chevalley-Eilenberg cohomology of the Lie algebra g. Finally, we show that for Lie algebras with K\"ahler structure we obtain a strongly positive universal deformation of *-algebras by using a Wick-type deformation. This results in a positive Drinfel'd twist.