Realization of Poisson enveloping algebra

doi:10.1007/s11464-018-0708-x

Frontiers of Mathematics in China

2018, Vol. 13

Issue (4): 999-1011 https://doi.org/10.1007/s11464-018-0708-x

本期目录

Realization of Poisson enveloping algebra

Can ZHU(

), Yaxiu WANG

College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China

全文: PDF(179 KB)

Abstract：

For a Poisson algebra, the category of Poisson modules is equivalent to the module category of its Poisson enveloping algebra, where the Poisson enveloping algebra is an associative one. In this article, for a Poisson structure on a polynomial algebra S, we first construct a Poisson algebra R, then prove that the Poisson enveloping algebra of S is isomorphic to the specialization of the quantized universal enveloping algebra of R, and therefore, is a deformation quantization of R.

Key words： Poisson enveloping algebra quantized universal enveloping algebra deformation quantization

收稿日期: 2018-01-27 出版日期: 2018-08-14

Corresponding Author(s): Can ZHU

引用本文:

. [J]. Frontiers of Mathematics in China, 2018, 13(4): 999-1011.
Can ZHU, Yaxiu WANG. Realization of Poisson enveloping algebra. Front. Math. China, 2018, 13(4): 999-1011.

链接本文:

https://academic.hep.com.cn/fmc/CN/10.1007/s11464-018-0708-x
https://academic.hep.com.cn/fmc/CN/Y2018/V13/I4/999

1	Brown K A, Gordon I. Poisson orders, symplectic reflection algebras and representation theory. J Reine Angew Math, 2003, 559: 193–216 https://doi.org/10.1515/crll.2003.048
2	Calaque D, Felder G, Rossi C. Deformation quantization with generators and relations. J Algebra, 2011, 337: 1–12 https://doi.org/10.1016/j.jalgebra.2011.03.037
3	Dolgushev V A. The Van den Bergh duality and the modular symmetry of a Poisson variety. Selecta Math, 2009, 14: 199–228 https://doi.org/10.1007/s00029-008-0062-z
4	Huebschmann J. Poisson cohomology and quantization. J Reine Angew Math, 1990, 408: 57–113
5	Kontsevich M. Deformation quantization of Poisson manifolds. Lett Math Phys, 2003, 66: 157–216 https://doi.org/10.1023/B:MATH.0000027508.00421.bf
6	Lü J, Wang X, Zhuang G. Universal enveloping algebras of Poisson Hopf algebras. J Algebra, 2015, 426: 92–136 https://doi.org/10.1016/j.jalgebra.2014.12.010
7	Lü J, Wang X, Zhuang G. Universal enveloping algebras of Poisson Ore extensions. Proc Amer Math Soc, 2015, 143: 4633–4645 https://doi.org/10.1090/S0002-9939-2015-12631-7
8	Lü J, Wang X, Zhuang G. DG Poisson algebra and its universal enveloping algebra. Sci China Math, 2016, 59: 849–860 https://doi.org/10.1007/s11425-016-5127-4
9	Lü J, Wang X, Zhuang G. Homological unimodularity and Calabi-Yau condition for Poisson algebras. Lett Math Phys, 2017, 107: 1715–1740 https://doi.org/10.1007/s11005-017-0967-6
10	Oh S Q. Poisson enveloping algebras. Comm Algebra, 1999, 27: 2181–2186 https://doi.org/10.1080/00927879908826556
11	Oh S Q, Park C G, Shin Y Y. A Poincaré-Birkhoff-Witt theorem for Poisson enveloping algebras. Comm Algebra, 2002, 30: 4867–4887 https://doi.org/10.1081/AGB-120014673
12	Penkava M, Vanhaecke P. Deformation quantization of polynomial Poisson algebras. J Algebra, 2000, 227: 365–393 https://doi.org/10.1006/jabr.1999.8239
13	Rinehart G. Differential forms on general commutative algebras. Trans Amer Math Soc, 1963, 108: 195–222 https://doi.org/10.1090/S0002-9947-1963-0154906-3
14	Shoikhet B. Kontsevich formality and PBW algebras. arXiv: 0708.1634
15	Shoikhet B. The PBW property for associative algebras as an integrability conditions. Math Res Lett, 2014, 21: 1407–1434 https://doi.org/10.4310/MRL.2014.v21.n6.a11
16	Towers M. Poisson and Hochschild cohomology and the semiclassical limit. J Noncommut Geom, 2015, 9: 665–696 https://doi.org/10.4171/JNCG/204
17	Umirbaev U. Universal enveloping algebras and universal derivations of Poisson algebras. J Algebra, 2012, 354: 77–94 https://doi.org/10.1016/j.jalgebra.2012.01.003
18	Vancliff M. Primitive and Poisson spectra of twists of polynomial rings. Algebr Represent Theory, 1999, 3: 269–285 https://doi.org/10.1023/A:1009914728281
19	Weinstein A. The local structure of Poisson manifolds. J Differential Geom, 1982, 18: 523–557 https://doi.org/10.4310/jdg/1214437787

Viewed

Full text

Abstract

Cited

Shared

Discussed