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The Conjecture of Dixmier

迪克斯米尔猜想

基本信息

批准号：
EP/J009342/1
负责人：
V Bavula
金额：
$ 7.04万
依托单位：
University of Sheffield
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2012
资助国家：
英国
起止时间：
2012 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FJ009342%2F1
关键词：
Conjecture Dixmier

项目摘要

In Mathematics there are two old open problems: the JacobianConjecture (open since 1938) for the polynomial algebras in n variables and theConjecture of Dixmier (open since 1968) for the algebras A(n) ofpolynomial differential operators, the so-called Weyl algebras, that claims thatthe Weyl algebras behave like the finite fields. More precisely, every algebra endomorphism of the Weyl algebra is anautomorphism. In 1982, Bass, Connell and Wright proved that theConjecture of Dixmier implies the Jacobian Conjecture. In2005-07, Tsuchimoto, Belov-Kanel and Kontsevich proved that thesetwo conjectures are equivalent. The Weyl algebra A(n) is a subalgebra of the algebra I(n) of polynomial integro-differentialoperators. At the end of 2010, I proved that an an analogue of theConjecture of Dixmier holds for the algebra I(1) (V. Bavula, ``Ananalogue of the Conjecture of Dixmier is true for the algebra ofpolynomial integro-differential operators,'' Arxiv:math.RA:1011.3009), and conjectured that the same result is true for allalgebras I(n). The aim of this project is to prove this conjectureand as a result to have a progress on the Conjecture of Dixmier.Another goal of the project is to find the K-groups for thealgebras I(n) and to answer the question of whether or not theBott periodicity holds. The most interesting (and difficult) isthe case of the K(1)-groups for the algebras I(n) since it leadsto finding explicit generators for the automorphism groups of thealgebras I(n). The groups of automorphisms of the algebras I(n)are infinite dimensional algebraic groups. Little is known abouttheir structure in general. In the polynomial case there areseveral papers by Shafarevich (1966, 1981) and more recently byKambayashi (1996, 2003, 2004). We are going to obtaingeneralizations of these results for the Weyl algebras A(n) andI(n).

在数学中有两个古老的开放问题：针对n变量多项式代数的jacobian猜想（1938年开放）和针对多项式微分算子的代数A(n)的Dixmier猜想（1968年开放），即所谓的Weyl代数，该猜想声称Weyl代数的行为类似于有限域。更确切地说，Weyl代数的每一个代数自同构都是反自同构。1982年，Bass、Connell和Wright证明了Dixmier猜想包含雅可比猜想。2005-07年，土本、Belov-Kanel和Kontsevich证明了这两个猜想是等价的。Weyl代数A(n)是多项式积分微分算子代数I(n)的一个子代数。在2010年底，我证明了Dixmier猜想的一个类似对代数I(1)成立（V. Bavula，“Dixmier猜想的一个类似对多项式积分微分算子的代数成立”，Arxiv:math.RA:1011.3009），并推测同样的结果对所有代数I(n)成立。本课题的目的是为了证明这一猜想，从而在迪克米耶猜想上取得进展。该项目的另一个目标是找到代数I(n)的k群，并回答博特周期性是否成立的问题。最有趣（也是最难）的是代数I(n)的K(1)群的情况，因为它导致找到代数I(n)的自同构群的显式生成器。代数I(n)的自同构群是无限维代数群。人们对它们的总体结构知之甚少。在多项式情况下，有Shafarevich（1966, 1981）和kambayashi（1996, 2003, 2004）的几篇论文。我们将得到Weyl代数A(n)和(n)的这些结果的推广。