Polynomial Automorphisms, Deformation Quantization and Some Applications on Noncommutative Algebras

Wenchao Zhang, Roman Yavich, Alexei Belov-Kanel, Farrokh Razavinia, Andrey Elishev, Jietai Yu

Research output: Contribution to journalReview articlepeer-review

Abstract

This paper surveys results concerning the quantization approach to the Jacobian Conjecture and related topics on noncommutative algebras. We start with a brief review of the paper and its motivations. The first section deals with the approximation by tame automorphisms and the Belov–Kontsevich Conjecture. The second section provides quantization proof of Bergman’s centralizer theorem which has not been revisited for almost 50 years and formulates several related centralizer problems. In the third section, we investigate a free algebra analogue of a classical theorem of Białynicki-Birula’s theorem and give a noncommutative version of this famous theorem. Additionally, we consider positive-root torus actions and obtain the linearity property analogous to the Białynicki-Birula theorem. In the last sections, we introduce Feigin’s homomorphisms and we see how they help us in proving our main and fundamental theorems on screening operators and in the construction of our lattice (Formula presented.) -algebras associated with (Formula presented.), which is by far the simplest known approach concerning constructing such algebras until now.

Original languageEnglish
Article number4214
JournalMathematics
Volume10
Issue number22
DOIs
StatePublished - Nov 2022

Bibliographical note

Publisher Copyright:
© 2022 by the authors.

Funding

This work is partially supported by the Russian Science Foundation (Grant No. 22-11-00177). The first author was partially supported by the Project of Guangdong Provincial Department of Education (Grant No. 2021ZDJS080) and the Professorial and Doctoral Scientific Research Foundation of Huizhou University (No. 2021JB022). F. Razavinia was partially supported and funded by the science foundation of Urmia University.

FundersFunder number
Project of Guangdong Provincial Department of Education2021ZDJS080
Huizhou University2021JB022
Russian Science Foundation22-11-00177
science foundation of Urmia University

    Keywords

    • Feigin’s homomorphisms
    • Lattice W-algebras
    • Weyl algebra
    • centralizers
    • deformation quantization
    • generic matrices
    • polynomial automorphisms
    • quantum groups
    • torus actions

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