The inversion formula for automorphisms of the Weyl algebras and polynomial algebras
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  • 作者：V.V. Bavula
  • 刊名：Journal of Pure and Applied Algebra
  • 出版年：2007
  • 出版时间：July 2007
  • 年：2007
  • 卷：210
  • 期：1
  • 页码：147-159
  • 全文大小：307 K
文摘
Let An be the nth Weyl algebra and Pm be a polynomial algebra in m variables over a field K of characteristic zero. The following characterization of the algebras {AnPm} is proved: an algebra A admits a finite set δ1,…,δs of commuting locally nilpotent derivations with generic kernels and iff AAnPm for some n and m with 2n+m=s, and vice versa. The inversion formula for automorphisms of the algebra AnPm (and for ) 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 , then (the proof is algebro-geometric). We extend this result (using [non-holonomic] -modules): given , then . Any automorphism 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 .

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:AnPm→AnPm an algebra automorphism provided σ(Pm)Pm and ? (Pm=K[x1,…,xm]). 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]])].

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