Lifting of polynomial symplectomorphisms and deformation quantization

ABSTRACT

We study the problem of lifting of polynomial symplectomorphisms in characteristic zero to automorphisms of the Weyl algebra by means of approximation by tame automorphisms. In 1983, Anick proved the fundamental result on approximation of polynomial automorphisms. We obtain similar approximation theorems for symplectomorphisms and Weyl algebra authomorphisms. We then formulate the lifting problem. More precisely, we prove the possibility of lifting of a symplectomorphism to an automorphism of the power series completion of the Weyl algebra of the corresponding rank. The lifting problem has its origins in the context of deformation quantization of the affine space and is closely related to several major open problems in algebraic geometry and ring theory.

This paper is a continuation of the study [19].

KEYWORDS:

• Jacobian conjecture
• polynomial automorphisms and symplectomorphisms
• tame and wild automorphisms
• quantization

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 14R10
• 14R15

Notes

¹We set deg x_i = 1.

²For W_n the degree is well defined, but the height depends on the ordering of the generators.

³Evidently, no loss of generality results from such explicit labelling.