Polynomial Symplectomorphisms and Bergman's Centralizer Theorem with Deformation Quantization​

Quantization ideas provides a new vision in some classical areas in algebra including polynomial automorphisms and Jacobian conjecture. In the first part of this talk, I will introduce lifting property of polynomial symplectomorphisms in characteristic zero to automorphisms of the Weyl algebra by means of approximation by tame automorphisms. We utilize and reprove D. Anick's fundamental result on approximation of polynomial automorphisms, adapt it to the case of symplectomorphisms, and formulate the lifting problem. In the second part, I will present an elegant reproof of famous Bergman's Centralizer Theorem which is also discovered by using deformation quantization and generic matrix.