Polynomial Automorphisms, Deformation Quantization and Some Applications on Noncommutative Algebras

Wenchao Zhang (), Roman Yavich, Alexei Belov-Kanel, Farrokh Razavinia, Andrey Elishev and Jietai Yu
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Wenchao Zhang: School of Mathematics and Statistics, Huizhou University, Huizhou 516007, China
Roman Yavich: Department of Mathematics, Ariel University, Ariel 4070000, Israel
Alexei Belov-Kanel: Department of Mathematics, Bar-Ilan University, Ramat Gan 5290002, Israel
Farrokh Razavinia: Department of Physics, Urmia University, Urmia 5756151818, West Azerbaijan Province, Iran
Andrey Elishev: Department of Discrete Mathematics, Moscow Institute of Physics and Technology, Dolgoprudny 141700, Moscow Region, Russia
Jietai Yu: College of Mathematics and Statistics, Shenzhen University, Shenzhen 518061, China

Mathematics, 2022, vol. 10, issue 22, 1-36

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 W n -algebras associated with sl n , which is by far the simplest known approach concerning constructing such algebras until now.

Keywords: deformation quantization; polynomial automorphisms; generic matrices; centralizers; torus actions; Weyl algebra; Lattice W-algebras; quantum groups; Feigin’s homomorphisms (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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