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A Universal Construction of Universal Deformation Formulas, Drinfeld Twists and their Positivity

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 Added by Chiara Esposito
 Publication date 2016
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and research's language is English




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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 Drinfeld theorem and to obtain a concrete formula for Drinfeld 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 Kahler structure we obtain a strongly positive universal deformation of *-algebras by using a Wick-type deformation. This results in a positive Drinfeld twist.



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We introduce a general theory of twisting algebraic structures based on actions of a bialgebra. These twists are closely related to algebraic deformations and also to the theory of quasi-triangular bialgebras. In particular, a deformation produced from a universal deformation formula (UDF) is a special case of a twist. The most familiar example of a deformation produced from a UDF is perhaps the Moyal product which (locally) is the canonical quantization of the algebra of functions on a symplectic manifold in the direction of the Poisson bracket. In this case, the derivations comprising the Poisson bracket mutually commute and so this quantization is essentially obtained by exponentiating this bracket. For more general Poisson manifolds, this formula is not applicable since the associated derivations may no longer commute. We provide here generalizations of the Moyal formula which (locally) give canonical quantizations of various Poisson manifolds. Specifically, whenever a certain central extension of a Heisenberg Lie group acts on a manifold, we obtain a quantization of its algebra of functions in the direction of a suitable Poisson bracket obtained from noncommuting derivations.
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