Czechoslovak Mathematical Journal, online first, 4 pp.

A note on Poisson derivations

Jiantao Li

Received November 2, 2016.   First published May 9, 2017.

Jiantao Li, School of Mathematics, Liaoning university, No. 66 Chongshan Middle Road, 110036 Shenyang, Huanggu, Liaoning, China, e-mail:

Abstract: Free Poisson algebras are very closely connected with polynomial algebras, and the Poisson brackets are used to solve many problems in affine algebraic geometry. In this note, we study Poisson derivations on the symplectic Poisson algebra, and give a connection between the Jacobian conjecture with derivations on the symplectic Poisson algebra.

Keywords: Poisson algebra; derivation; Jacobian conjecture

Classification (MSC 2010): 13N15, 17B63

DOI: 10.21136/CMJ.2017.0574-16

Full text available as PDF.

  [1] L. Makar-Limanov, I. Shestakov: Polynomial and Poisson dependence in free Poisson algebras and free Poisson fields. J. Algebra 349 (2012), 372-379. DOI 10.1016/j.jalgebra.2011.08.008 | MR 2853644 | Zbl 1259.17021
  [2] L. Makar-Limanov, U. Turusbekova, U. Umirbaev: Automorphisms and derivations of free Poisson algebras in two variables. J. Algebra 322 (2009), 3318-3330. DOI 10.1016/j.jalgebra.2008.01.005 | MR 2567422 | Zbl 1233.17016
  [3] L. Makar-Limanov, U. Umirbaev: Centralizers in free Poisson algebras. Proc. Amer. Math. Soc. 135 (2007), 1969-1975. DOI 10.1090/S0002-9939-07-08678-9 | MR 2299468 | Zbl 1158.17006
  [4] L. Makar-Limanov, U. Umirbaev: The Freiheitssatz for Poisson algebras. J. Algebra 328 (2011), 495-503. DOI 10.1016/j.jalgebra.2010.08.015 | MR 2745580 | Zbl 1285.17007
  [5] A. Nowicki: Polynomial Derivations and Their Rings of Constants. Uniwersytet Mikołaja Kopernika, Toruń (1994). MR 2553232 | Zbl 1236.13023
  [6] I. P. Shestakov, U. U. Umirbaev: Poisson brackets and two-generated subalgebras of rings of polynomials. J. Am. Math. Soc. 17 (2004), 181-196. DOI 10.1090/S0894-0347-03-00438-7 | MR 2015333 | Zbl 1044.17014
  [7] I. P. Shestakov, U. U. Umirbaev: The tame and the wild automorphisms of polynomial rings in three variables. J. Am. Math. Soc. 17 (2004), 197-227. DOI 10.1090/S0894-0347-03-00440-5 | MR 2015334 | Zbl 1056.14085
  [8] U. Umirbaev: Universal enveloping algebras and universal derivations of Poisson algebras. J. Algebra 354 (2012), 77-94. DOI 10.1016/j.jalgebra.2012.01.003 | MR 2879224 | Zbl 1270.17013
  [9] A. van den Essen: Polynomial Automorphisms and the Jacobian Conjecture. Progress in Mathematics 190, Birkhäuser, Basel (2000). DOI 10.1007/978-3-0348-8440-2 | MR 1790619 | Zbl 0962.14037

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