The inversion formula for automorphisms of the Weyl algebras and polynomial algebras

Abstract

Let A n be the n th Weyl algebra and P m be a polynomial algebra in m variables over a field K of characteristic zero. The following characterization of the algebras { A n ⊗ P m } is proved: an algebra A admits a finite set δ 1 , … , δ s of commuting locally nilpotent derivations with generic kernels and ∩ i = 1 s ker ( δ i ) = K iff A ≃ A n ⊗ P m for some n and m with 2 n + m = s , and vice versa. The inversion formula for automorphisms of the algebra A n ⊗ P m (and for P ̂ m ≔ K 〚 x 1 , … , x m 〛 ) has been found (giving a new inversion formula even for polynomials). Recall that (see [H. Bass, E.H. Connell, D. Wright, The Jacobian Conjecture: Reduction of degree and formal expansion of the inverse, Bull. Amer. Math. Soc. (New Series) 7 (1982) 287–330]) given σ ∈ Aut K ( P m ) , then deg σ − 1 ≤ ( deg σ ) m − 1 (the proof is algebro-geometric). We extend this result (using [non-holonomic] D - modules): given σ ∈ Aut K ( A n ⊗ P m ) , then deg σ − 1 ≤ ( deg σ ) 2 n + m − 1 . Any automorphism σ ∈ Aut K ( P m ) is determined by its face polynomials [J.H. McKay, S.S.-S. Wang, On the inversion formula for two polynomials in two variables, J. Pure Appl. Algebra 52 (1988) 102–119], a similar result is proved for σ ∈ Aut K ( A n ⊗ P m ) . One can amalgamate two old open problems ( the Jacobian Conjecture and the Dixmier Problem, see [J. Dixmier, Sur les algèbres de Weyl, Bull. Soc. Math. France 96 (1968) 209–242. [6]] problem 1) into a single question, ( JD): is a K - algebra endomorphism σ : A n ⊗ P m → A n ⊗ P m an algebra automorphism provided σ ( P m ) ⊆ P m and det ( ∂ σ ( x i ) ∂ x j ) ∈ K ∗ ≔ K ∖ { 0 } ? ( P m = K [ x 1 , … , x m ] ) . It follows immediately from the inversion formula that this question has an affirmative answer iff both conjectures have (see below) [ iff one of the conjectures has a positive answer (as follows from the recent papers [Y. Tsuchimoto, Endomorphisms of Weyl algebra and p -curvatures, Osaka J. Math. 42(2) (2005) 435–452. [10]] and [A. Belov-Kanel, M. Kontsevich, The Jacobian conjecture is stably equivalent to the Dixmier Conjecture. ArXiv:math.RA/0512171. [5]])].