×

Obstructions for twist star products. (English) Zbl 1392.53082

This paper studies obstructions for a given star product to be induced by a twist. First, the authors recall the main notions and results about Drinfel’d twist and classical \(r\)-matrix. Then, they prove that not every Poisson structure is induced by a classical \(r\)-matrix. Counterexamples are obtained by symplectic manifolds which are not homogeneous and by higher-genus Pretzel surfaces or by the sphere.

MSC:

53D05 Symplectic manifolds (general theory)
53D55 Deformation quantization, star products
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Aschieri, P., Schenkel, A.: Noncommutative connections on bimodules and Drinfel’d twist deformation. Adv. Theor. Math. Phys. 18(3), 513-612 (2014) · Zbl 1317.14008 · doi:10.4310/ATMP.2014.v18.n3.a1
[2] Bayen, F., Flato, M., Frønsdal, C., Lichnerowicz, A., Sternheimer, D.: Deformation theory and quantization. Ann. Phys. 111, 61-151 (1978) · Zbl 0377.53024 · doi:10.1016/0003-4916(78)90224-5
[3] Connes, A.: Noncommutative Geometry. Academic Press, San Diego (1994) · Zbl 0818.46076
[4] Drinfel’d, V.G.: On constant quasiclassical solutions of the Yang-Baxter quantum equation. Sov. Math. Dokl. 28, 667-671 (1983) · Zbl 0553.58038
[5] Drinfeld, V.G.: Quantum groups. J. Sov. Math. 41, 898-918 (1988) · Zbl 0641.16006 · doi:10.1007/BF01247086
[6] Etingof, P., Schiffmann, O.: Lectures on Quantum Groups. International Press, Boston (1998) · Zbl 1105.17300
[7] Giaquinto, A., Zhang, J.J.: Bialgebra actions, twists, and universal deformation formulas. J. Pure Appl. Algebra 128(2), 133-152 (1998) · Zbl 0938.17015 · doi:10.1016/S0022-4049(97)00041-8
[8] Montgomery, D., Samelson, H.: Transformation groups of spheres. Ann. Math. 44(3), 454 (1942) · Zbl 0063.04077 · doi:10.2307/1968975
[9] Mostow, G.D.: The extensibility of local Lie groups of transformations and groups on surfaces. Ann. Math. (2) 52, 606-636 (1950) · Zbl 0040.15204 · doi:10.2307/1969437
[10] Mostow, G.D.: A structure theorem for homogeneous spaces. Geom. Dedic. 114, 87-102 (2005) · Zbl 1086.57024 · doi:10.1007/s10711-004-1675-9
[11] Onishchik, A.: On lie groups transitive on compact manifolds II. Math. USSR-Sbornik 3(3), 373 (1967) · Zbl 0198.28903 · doi:10.1070/SM1967v003n03ABEH002750
[12] Onishchik, A.: On lie groups transitive on compact manifolds III. Math. USSR-Sbornik 4(2), 233 (1968) · Zbl 0198.29001 · doi:10.1070/SM1968v004n02ABEH002794
[13] Onishchik, A.: Lie Groups and Lie Algebras I. Encyclopaedia of Mathematical Sciences. Springer, Berlin (1993) · Zbl 0777.00023 · doi:10.1007/978-3-642-57999-8
[14] Palais, R.S.: A global formulation of the Lie theory of transformation groups, 22 (1957) · Zbl 0178.26502
[15] Waldmann, S.: Poisson-Geometrie und Deformationsquantisierung. Eine Einführung. Springer, Heidelberg (2007) · Zbl 1139.53001
[16] Weber, T.: Star Products that can not be induced by Drinfel’d Twists. master thesis, University of Würzburg, Würzburg, Germany (2016)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.