Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 026, 23 pages      arXiv:1605.01376      https://doi.org/10.3842/SIGMA.2018.026

Hopf Algebroid Twists for Deformation Quantization of Linear Poisson Structures

Stjepan Meljanac a and Zoran Škoda bc
a) Theoretical Physics Division, Institute Rudjer Bošković, Bijenička cesta 54, P.O. Box 180, HR-10002 Zagreb, Croatia
b) Faculty of Science, University of Hradec Králové, Rokitanského 62, Hradec Králové, Czech Republic
c) University of Zadar, Department of Teachers' Education, Franje Tudjmana 24, 23000 Zadar, Croatia

Received May 24, 2017, in final form March 13, 2018; Published online March 25, 2018

Abstract
In our earlier article [Lett. Math. Phys. 107 (2017), 475-503], we explicitly described a topological Hopf algebroid playing the role of the noncommutative phase space of Lie algebra type. Ping Xu has shown that every deformation quantization leads to a Drinfeld twist of the associative bialgebroid of $h$-adic series of differential operators on a fixed Poisson manifold. In the case of linear Poisson structures, the twisted bialgebroid essentially coincides with our construction. Using our explicit description of the Hopf algebroid, we compute the corresponding Drinfeld twist explicitly as a product of two exponential expressions.

Key words: deformation quantization; Hopf algebroid; noncommutative phase space; Drinfeld twist; linear Poisson structure.

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