RESEARCH ARTICLE

Realization of Poisson enveloping algebra

  • Can ZHU ,
  • Yaxiu WANG
Expand
  • College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China

Received date: 27 Jan 2018

Accepted date: 28 May 2018

Published date: 14 Aug 2018

Copyright

2018 ©后内容,如Higher Education Press and Springer-Verlag Berlin Heidelberg

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.

Cite this article

Can ZHU , Yaxiu WANG . Realization of Poisson enveloping algebra[J]. Frontiers of Mathematics in China, 2018 , 13(4) : 999 -1011 . DOI: 10.1007/s11464-018-0708-x

1
Brown K A, Gordon I. Poisson orders, symplectic reflection algebras and representation theory. J Reine Angew Math, 2003, 559: 193–216

DOI

2
Calaque D, Felder G, Rossi C. Deformation quantization with generators and relations. J Algebra, 2011, 337: 1–12

DOI

3
Dolgushev V A. The Van den Bergh duality and the modular symmetry of a Poisson variety. Selecta Math, 2009, 14: 199–228

DOI

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

DOI

6
Lü J, Wang X, Zhuang G. Universal enveloping algebras of Poisson Hopf algebras. J Algebra, 2015, 426: 92–136

DOI

7
Lü J, Wang X, Zhuang G. Universal enveloping algebras of Poisson Ore extensions. Proc Amer Math Soc, 2015, 143: 4633–4645

DOI

8
Lü J, Wang X, Zhuang G. DG Poisson algebra and its universal enveloping algebra. Sci China Math, 2016, 59: 849–860

DOI

9
Lü J, Wang X, Zhuang G. Homological unimodularity and Calabi-Yau condition for Poisson algebras. Lett Math Phys, 2017, 107: 1715–1740

DOI

10
Oh S Q. Poisson enveloping algebras. Comm Algebra, 1999, 27: 2181–2186

DOI

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

DOI

12
Penkava M, Vanhaecke P. Deformation quantization of polynomial Poisson algebras. J Algebra, 2000, 227: 365–393

DOI

13
Rinehart G. Differential forms on general commutative algebras. Trans Amer Math Soc, 1963, 108: 195–222

DOI

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

DOI

16
Towers M. Poisson and Hochschild cohomology and the semiclassical limit. J Noncommut Geom, 2015, 9: 665–696

DOI

17
Umirbaev U. Universal enveloping algebras and universal derivations of Poisson algebras. J Algebra, 2012, 354: 77–94

DOI

18
Vancliff M. Primitive and Poisson spectra of twists of polynomial rings. Algebr Represent Theory, 1999, 3: 269–285

DOI

19
Weinstein A. The local structure of Poisson manifolds. J Differential Geom, 1982, 18: 523–557

DOI

Outlines

/