The inversion formula for automorphisms of the Weyl algebras and polynomial algebras
The inversion formula for automorphisms of the Weyl algebras and polynomial algebras
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Weyl代数和多项式代数自同构的反演公式
DOI：
10.1016/j.jpaa.2006.09.002
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发表时间：
2005
影响因子：
0.8
通讯作者：
V. Bavula
中科院分区：
文献类型：
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作者：
V. Bavula
设Anbe是特征为零的域K上的n阶Weyl代数，Pmb是m个变量的多项式代数。证明了代数{an⊗Pm}的如下刻划：代数A可容纳一个有限集δ1，…，δsof交换局部幂零导子，∩i=1sker(δi)=KiffA≃an⊗Pm，2n+m=S，反之亦然。代数An⊗Pm(和P̂m≔K[x1，…)的自同构的求逆公式，XM])得到了(即使对于多项式也给出了一个新的求逆公式)。回想一下(参见[H.Bass，E.H.Connell，D.Wright，The Jacobian猜想：降阶和逆数的形式展开，Bull。阿默。数学课。SoC。(新系列)7(1982)287-330])给定σ∈autK(PM)，则σ−1≤(degσ)m−1(证明是代数几何的)。我们推广了这一结果(使用[非完整]D-模)：给定σ∈AutK(an⊗Pm)，则σ−1≤(degσ)2n+m−1.任何自同构σ∈AutK(Pm)由它的面多项式[J.H.McKay，S.S.-S.Wang，关于两个变量多项式的逆公式J.Pure Appl.代数52(1988)102-119]，对σ∈AutK(an⊗PM)也证明了类似的结果。人们可以将两个老的公开问题(雅可比猜想和迪克斯米尔问题，见J·迪克西米尔，Sur les Algèbres de Weyl，Bull)合并在一起。SoC。数学课。法国96(1968)209-242。]问题1)变成一个问题，(JD)：AK-代数自同态σ：an⊗pm→an⊗Pman代数自同构提供了σ(Pm)⊆Pmanddet(∂σ(Xi)∂xj)∈K∗≔K∖{0}？(Pm=K[x1，…，XM]))。从倒置公式可以立即得出这个问题有肯定答案当且仅当两个猜想都有(见下文)[当其中一个猜想有肯定答案(如最近的论文[Y.Tsuhimoto，Weyl代数的自同态和p-曲率，Osaka J.Math.42(2)(2005)435-452]和[A.Belov-Kanel，M.Kontsevich，Jacobian猜想稳定地等价于Dixmier型猜想。。])]。
Let Anbe the nth Weyl algebra and Pmbe a polynomial algebra in m variables over a field K of characteristic zero. The following characterization of the algebras {An⊗Pm} is proved: an algebraAadmits a finite setδ1,…,δsof commuting locally nilpotent derivations with generic kernels and∩i=1sker(δi)=KiffA≃An⊗Pmfor somenandmwith2n+m=s, and vice versa. The inversion formula for automorphisms of the algebra An⊗Pm(and for P̂m≔K[x1,…,xm]) 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]) givenσ∈AutK(Pm), thendegσ−1≤(degσ)m−1(the proof is algebro-geometric). We extend this result (using [non-holonomic] D-modules): givenσ∈AutK(An⊗Pm), thendegσ−1≤(degσ)2n+m−1. Any automorphism σ∈AutK(Pm) 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∈AutK(An⊗Pm). 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. ] problem 1) into a single question, (JD): is aK-algebra endomorphismσ:An⊗Pm→An⊗Pman algebra automorphism providedσ(Pm)⊆Pmanddet(∂σ(xi)∂xj)∈K∗≔K∖{0}? (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. ] and [A. Belov-Kanel, M. Kontsevich, The Jacobian conjecture is stably equivalent to the Dixmier Conjecture. . ])].