About Dixmier's conjecture

作者：Moskowicz Vered^*

来源：Journal of Algebra and Its Applications, 2015, 14(10): 1550140.

DOI：10.1142/S0219498815501406

• Weyl algebra
• Dixmier conjecture
• rings with involution
• Jacobian conjecture

The well-known Dixmier conjecture [5] asks if every algebra endomorphism of the first Weyl algebra over a characteristic zero field is an automorphism. We bring a hopefully easier to solve conjecture, called the gamma, delta conjecture, and show that it is equivalent to the Dixmier conjecture. In the group generated by automorphisms and anti-automorphisms of A(1), all the involutions belong to one conjugacy class, hence:
Every involutive endomorphism from (A(1),gamma) to (A(1), delta) is an automorphism (gamma and delta are two involutions on A(1)).
Given an endomorphism f of A(1) (not necessarily an involutive endomorphism), if one of f(X), f(Y) is symmetric or skew-symmetric (with respect to any involution on A(1)), then f is an automorphism.

• 出版日期2015-12