A universal construction of universal deformation formulas, Drinfeld twists and their positivity

Chiara Esposito, Jonas Schnitzer and Stefan Waldmann

Vol. 291 (2017), No. 2, 319–358

DOI: 10.2140/pjm.2017.291.319

Abstract

We provide an explicit construction of star products on $\mathcal{U}\left(\mathfrak{g}\right)$-module algebras by using the Fedosov approach. This allows us to give a constructive proof to Drinfeld’s theorem and to obtain a concrete formula for Drinfeld twists. We prove that the equivalence classes of twists are in one-to-one correspondence with the second Chevalley–Eilenberg cohomology of the Lie algebra $\mathfrak{g}$. Finally, we show that for Lie algebras with Kähler structure we obtain a strongly positive universal deformation of ${}^{\ast }$-algebras by using a Wick-type deformation. This results in a positive Drinfeld twist.

Keywords

Star Product, Drinfeld twist, universal deformation formula

Mathematical Subject Classification 2010

Primary: 17B37

Secondary: 53D55, 81R50

Milestones

Received: 11 August 2016

Revised: 25 April 2017

Accepted: 9 May 2017

Published: 14 September 2017