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A universal construction of universal deformation formulas, Drinfeld twists and their positivity. (English) Zbl 1425.17019
Summary: We provide an explicit construction of star products on \(\mathcal{U}(\mathfrak{g})\)-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 \(^*\)-algebras by using a Wick-type deformation. This results in a positive Drinfeld twist.

MSC:
17B37 Quantum groups (quantized enveloping algebras) and related deformations
53D55 Deformation quantization, star products
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
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