Polynomial Automorphisms, Deformation Quantization and Some Applications on Noncommutative Algebras

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