[论文解读] Deformation cohomology of algebraic and geometric structures

J.-F. Pommaret|arXiv (Cornell University)|Jul 9, 2012

Advanced Topics in Algebra参考文献 16被引用 4

一句话总结

本文通过用 Vessiot 结构方程和 Spencer 算子替代传统的李代数方法，提出了一套统一的变形上同调框架，用于代数与几何结构，实现了对无限维李代数丛的计算机代数系统化应用。关键贡献是基于 Vessiot 形式化的新型上同调序列，该序列推广了 Chevalley-Eilenberg 上同调，并适用于黎曼、辛和接触几何等结构。

ABSTRACT

In 1953, the physicists E. Inonü and E.P. Wigner introduced the concept of deformation of a Lie algebra by claiming that the limit $1/c ightarrow 0$, when c is the speed of light, of the composition law $(u,v) ightarrow (u+v)/(1+(uv/c^2))$ of speeds in special relativity (Poincaré group) should produce the composition law $(u,v) ightarrow u + v $ used in classical mechanics (Galilée group). However, the dimensionless composition law $(u'=u/c,v'=v/c) ightarrow (u'+v')/(1+u'v')$ does not contain any longer a perturbation parameter. Nevertheless, this idea brought the birth of the " deformation theory of algebraic structures", culminating in the use of the Chevalley-Eilenberg cohomology of Lie algebras and one of the first applications of computer algebra in the seventies. One may also notice that the main idea of general relativity is to deform the Minkowski metric of space-time by means of the small dimensionless parameter $ϕ/c^2$ where $ϕ=GM/r$ is the gravitational potential at a distance r of a central attractive mass M with gravitational constant G. A few years later, a " deformation theory of geometric structures " on manifolds of dimension n was introduced and one may quote riemannian, symplectic or complex analytic structures. Though often conjectured, the link between the two approaches has never been exhibited and the aim of this paper is to provide the solution of this problem by new methods. The key tool is made by the " Vessiot structure equations " (1903) for Lie groups or Lie pseudogroups of transformations, which, contrary to the " Cartan structure equations ", are still unknown today and contain " structure constants " which, like in the case of constant riemannian curvature, have in general nothing to do with any Lie algebra. The main idea is then to introduce the purely differential Janet sequence $0 ightarrow Θ ightarrow T ightarrow F_0 ightarrow F_1 ightarrow ... ightarrow F_n ightarrow 0$ as a resolution of the sheaf $Θ\subset T$ of infinitesimal transformations and to induce a purely algebraic " deformation sequence " with finite dimensional vector spaces and linear maps, even if $Θ$ is infinite dimensional. The infinitesimal equivalence problem for geometric structures has to do with the local exactness at $ F_0 $ of the Janet sequence while the deformation problem for algebraic structures has to do with the exactness of the deformation sequence at the invariant sections of $F_1 $, that is ONE STEP FURTHER ON in the sequence and this unexpected result explains why the many tentatives previously quoted have not been successful. Finally, we emphasize through examples the part that could be played by computer algebra in any explicit computation.

研究动机与目标

• 解决李代数变形理论与流形上几何结构之间长期存在的理论鸿沟。
• 基于 Vessiot 结构方程，为几何对象（如黎曼、辛、接触结构）的变形建立严谨的上同调框架。
• 证明 Chevalley-Eilenberg 上同调是源自 Spencer 算子与 Vessiot 形式化的更广泛上同调序列的特例。
• 通过使用有限维线性序列，即使在无限维李代数丛的情形下，也能系统化地应用计算机代数。
• 通过用 Vessiot 结构方程替代 Cartan 结构方程，阐明其自 1903 年以来被忽视的基础性作用，从而在变形程序中实现理论澄清。

提出的方法

• 用 Vessiot 结构方程替代 Cartan 结构方程，后者是喷射丛与自然丛几何的内在结构。
• 利用 Spencer 算子定义一个规范的线性 Janet 序列，该序列在有限维向量空间上诱导出一个上同调序列。
• 基于 Vessiot 上同调构建一个变形复形，其中上同调群仅依赖于 Vessiot 方程中的结构常数。
• 将该理论应用于主齐性空间，以恢复 Chevalley-Eilenberg 上同调作为特例。
• 利用 RWTH Aachen 开发的符号计算工具（如 Barakat 和 Lorenz 的工作）实现上同调序列的算法化处理。
• 通过非线性 Spencer 序列与喷射群胚形式化，系统地处理非交换与非线性几何结构。

实验结果

研究问题

• RQ1 如何通过统一框架将李代数的变形上同调推广至流形上的几何结构？
• RQ2 Vessiot 结构方程在变形理论中取代 Cartan 结构方程所起的作用是什么？
• RQ3 Spencer 算子如何实现对无限维李代数丛的有限维上同调序列的构造？
• RQ4 在李群情形下，新上同调序列如何恢复经典形式的 Chevalley-Eilenberg 上同调？
• RQ5 计算机代数系统能否有效计算复杂几何结构（如接触结构或单连通接触结构）的变形上同调？

主要发现

• Vessiot 结构方程为变形理论中的 Cartan 结构方程提供了基础性替代，解决了长期存在的概念性鸿沟。
• 即使在不存在底层李代数的场合（如常曲率黎曼流形），几何结构的变形上同调仍完全由 Vessiot 方程中的结构常数决定。
• 源自 Spencer 算子与 Janet 理论的新上同调序列是有限维且可计算的，从而可通过符号计算实现算法化处理。
• 当结构为李群的主齐性空间时，Chevalley-Eilenberg 上同调作为特例自然出现。
• 该框架成功将变形理论推广至无限维李代数丛，例如与接触结构及单连通接触结构相关的结构。
• 该理论在李代数的变形与几何结构的变形之间建立了概念上的桥梁，解决了 Spencer、Kodaira、Kuranishi 等人先前工作中未解决的猜想关联。