Vol. 291, No. 2, 2017

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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
Abstract

We provide an explicit construction of star products on $\mathsc{U}\left(\mathfrak{g}\right)$-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