Poisson double extensions

  • Qi Lou
  • Sei-Qwon Oh
  • Shengqiang WangEmail author


A double Ore extension was introduced by Zhang and Zhang (2008) to study a class of Artin-Shelter regular algebras. Here we give a definition of Poisson double extension which may be considered as an analogue of double Ore extension, and show that algebras in a class of double Ore extensions are deformation quantizations of Poisson double extensions. We also investigate the modular derivations of Poisson double extensions and the relationship between Poisson double extensions and iterated Poisson polynomial extensions. Results are illustrated by examples.


Poisson double extension semiclassical limit deformation quantization 


17B63 16S36 


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The first author was supported by National Natural Science Foundation of China (Grant No. 11331006). The second author was supported by National Research Foundation of Korea (Grant No. NRF-2017R1A2B4008388), and he thanks the Korea Institute for Advanced Study for the warm hospitality during the preparation of this paper. The third author was supported by National Natural Science Foundation of China (Grant Nos. 11301180 and 11771085). All the authors thank the referees for their helpful suggestions and comments.


  1. 1.
    Artin M, Schelter W F. Graded algebras of global dimension 3. Adv Math, 1987, 66: 171–216MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bergman G M. The diamond lemma for ring theory. Adv Math, 1978, 29: 178–218MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bitoun T. The p-support of a holonomic D-module is Lagrangian, for p large enough. ArXiv:1012.4081, 2010Google Scholar
  4. 4.
    Brown K A, Goodearl K R. Lectures on Algebraic Quantum Groups. Advanced Courses in Mathematics. Basel-Boston-Berlin: Birkhäuser-Verlag, 2002CrossRefGoogle Scholar
  5. 5.
    Cavalho P A, Lopes S A, Matczuk J. Double Ore extensions versus iterated Ore extensions. Comm Algebra, 2011, 39: 2838–2848MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cho E-H, Oh S-Q. Semiclassical limits of Ore extensions and a Poisson generalized Weyl algebra. Lett Math Phys, 2016, 106: 997–1009MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Dolgushev V A. The Van den Bergh duality and the modular symmetry of a Poisson variety. Selecta Math, 2009, 14: 199–228MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Goodearl K R. A Dixmier-Moeglin equivalence for Poisson algebras with torus actions. Contemp Math, 2006, 419: 131–154MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Huebschmann J. Duality for Lie-Rinehart algebras and the modular class. J Reine Angew Math, 1999, 510: 103–159MathSciNetzbMATHGoogle Scholar
  10. 10.
    Jordan D A, Oh S-Q. Poisson brackets and Poisson spectra in polynomial algebras. Contemp Math, 2012, 562: 169–187MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Jordan D A, Sasom N. Reversible skew Laurent polynomial rings and deformations of Poisson automorphisms. J Algebra Appl, 2009, 8: 733–757MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kontsevich M. Deformation quantization of Poisson manifolds I. ArXiv:q-alg/9709040, 1997zbMATHGoogle Scholar
  13. 13.
    Launois S, Lecoutre C. A quadratic Poisson Gel’fand-Kirillov problem in prime characteristic. Trans Amer Math Soc, 2016, 368: 755–785MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Luo J, Wang S Q, Wu Q S. Twisted Poincaré duality between Poisson homology and Poisson cohomology. J Algebra, 2015, 442: 484–505MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Lü J, Oh S-Q, Wang X, et al. Enveloping algebras of double Poisson-Ore extensions. Comm Algebra, 2018, 46: 4891–4904MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Lü J, Wang X, Zhuang G. Universal enveloping algebras of Poisson Ore extensions. Proc Amer Math Soc, 2015, 143: 4633–4645MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Oh S-Q. Poisson enveloping algebras. Comm Algebra, 1999, 27: 2181–2186MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Oh S-Q. Poisson polynomial rings. Comm Algebra, 2006, 34: 1265–1277MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Oh S-Q. A natural map from a quantized space onto its semiclassical limit and a multi-parameter Poisson Weyl algebra. Comm Algebra, 2017, 45: 60–75MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Rowen L H. Graduate Algebra: Commutative View. Graduate Studies in Mathematics, vol. 73. Providence: Amer Math Soc, 2006zbMATHGoogle Scholar
  21. 21.
    Van den Bergh M. Noncommutative homology of some three-dimensional quantum spaces. K-Theory, 1994, 8: 213–230MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Van den Bergh M. On involutivity of p-support. Int Math Res Notes IMRN, 2015, 15: 6295–6304MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Wang S Q. The modular derivations for extensions of Poisson algebras. Front Math China, 2017, 12: 209–218MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Zhang J J, Zhang J. Double Ore extensions. J Pure Appl Algebra, 2008, 212: 2668–2690MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Zhu C, Van Oystaeyen F, Zhang Y H. On (co)homology of Frobenius Poisson algebras. J K-Theory: K-Theory Appl Algebra Geom Topol, 2014, 14: 371–386MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Zhu C, Wang Y X. Realization of Poisson enveloping algebra. Front Math China, 2018, 13: 999–1011MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematical SciencesFudan UniversityShanghaiChina
  2. 2.Department of MathematicsChungnam National UniversityDaejeonRepublic of Korea
  3. 3.Department of MathematicsEast China University of Science and TechnologyShanghaiChina

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