Abstract
In this letter, first we give a decomposition for any Lie–Poisson structure \(\pi_{\mathfrak g}\) associated to the modular vector. In particular, \(\pi_{\mathfrak g}\) splits into two compatible Lie–Poisson structures if \({\rm dim}{\mathfrak g} \le 3\). As an application, we classified quadratic deformations of Lie– Poisson structures on \(\mathbb R^3\) up to linear diffeomorphisms.
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References
Cahen M. and Schwachhöfer L. (2004). Special symplectic connections and Poisson geometry. Lett. Math. Phys. 69: 115–137
Carinena J., Ibort A., Marmo G. and Perelomov A. (1994). On the geometry of Lie algebras and Poisson tensors. J. Phys. A Math. 27: 7425–7449
Chi Q.-S., Merkulov S. and Schwachhöfer L.J. (1996). On the existence of infinite series of exotic holonomies. Invent. Math. 126: 391–411
Crainic M. and Fernands R.L. (2004). Integrability of Poisson brackets. J. Diff. Geom. 66: 71–137
Donin J. and Makar L. (1998). Quantization of quadratic Poisson brackets on a polynomial algebra of three variables. J. Pure Appl. Algebra 129: 247–261
Dufour J.-P. and Haraki A. (1991). Rotationnels et structures de Poisson quadratiques. C. R. Acad. Sci. Paris, t. 312(Série I): 137–140
Dufour J.-P. and Zung N.-T. (2005). Poisson Structures and Their Normal Forms, PM242. Birkhauser, Boston
Grabowski J., Marmo G. and Perelomov A. (1993). Poisson structures: towards a classification. Mod. Phys. Lett A 8: 1719–1733
Koszul, J.-L.: Crochets de Schouten-Nijenhuis et cohomologei. Astérisque Soc. Math. de France,hors série 257–271 (1985)
Jacobson N. (1962). Lie Algbras. Dover Publications, Inc., New York
Liu Z.-J. and Xu P. (1992). On quadratic Poisson structures. Lett. Math. Phys. 26: 33–42
Manchon D., Msmoudi M. and Roux A. (2002). On quantization of quadratic Poisson structures. Commun. Math. Phys. 225: 121–130
Sheng Y.-H. (2007). Linear Poisson Strutures on \(\mathbb R^4\). J. Geom. Phys. 57: 2398–2410
Weinstein A. (1997). The modular automorphism group of a Poisson manifold. J. Geom. Phys. 26: 379–394
Weinstein A. (1998). Poisson geometry. Diff. Geom. Appl. 9: 213–238
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Research partially supported by NSF of China and the Research Project of “Nonlinear Science”.
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Lin, Q., Liu, Z. & Sheng, Y. Quadratic Deformations of Lie–Poisson Structures. Lett Math Phys 83, 217–229 (2008). https://doi.org/10.1007/s11005-008-0221-3
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DOI: https://doi.org/10.1007/s11005-008-0221-3