## Abstract

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 language | English |
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Pages (from-to) | 3926-3938 |

Number of pages | 13 |

Journal | Communications in Algebra |

Volume | 46 |

Issue number | 9 |

DOIs | |

State | Published - 2 Sep 2018 |

### Bibliographical note

Funding Information:This paper is supported by the Russian Science Foundation grant No. 17-11-01377.

Publisher Copyright:

© 2018, © 2018 Taylor & Francis.

## Keywords

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