Lifting of polynomial symplectomorphisms and deformation quantization

Alexei Kanel-Belov, Sergey Grigoriev, Andrey Elishev, Jie Tai Yu, Wenchao Zhang

Research output: Contribution to journalArticlepeer-review


We study the problem of lifting of polynomial symplectomorphisms in characteristic zero to automorphisms of the Weyl algebra by means of approximation by tame automorphisms. In 1983, Anick proved the fundamental result on approximation of polynomial automorphisms. We obtain similar approximation theorems for symplectomorphisms and Weyl algebra authomorphisms. We then formulate the lifting problem. More precisely, we prove the possibility of lifting of a symplectomorphism to an automorphism of the power series completion of the Weyl algebra of the corresponding rank. The lifting problem has its origins in the context of deformation quantization of the affine space and is closely related to several major open problems in algebraic geometry and ring theory. This paper is a continuation of the study [19].

Original languageEnglish
Pages (from-to)3926-3938
Number of pages13
JournalCommunications in Algebra
Issue number9
StatePublished - 2 Sep 2018


  • Jacobian conjecture
  • polynomial automorphisms and symplectomorphisms
  • quantization
  • tame and wild automorphisms

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory


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