代数和几何结构的变形上同调
Deformation cohomology of algebraic and geometric structures
摘要：
1953年，物理学家E.伊诺努和E. P.维格纳通过声称狭义相对论（庞加莱群）中速度合成律$(u,v) \rightarrow (u+v)/(1+(uv/c^2))$的极限$1/c \rightarrow 0$（当c是光速时）应该产生经典力学（伽利略群）中使用的合成律$(u,v) \rightarrow u + v $来引入李代数的变形概念。然而，无因次组成定律$(u'=u/c,v'=v/c) \rightarrow (u'+v')/(1+u'v')$不再包含扰动参数。然而，这个想法带来了“代数结构的变形理论”的诞生，最终在李代数的Chevalley-Eilenberg上同调的使用和计算机代数在七十年代的第一个应用中达到高潮。我们还可能注意到，广义相对论的主要思想是通过小的无量纲参数$ϕ/c^2$来变形时空的闵可夫斯基度规，其中$ϕ=GM/r$是距离r为中心吸引质量M且引力常数G的引力势。几年后，一个“变形理论的几何结构“的流形上的维数n的介绍，人们可以引用黎曼，辛或复杂的分析结构。虽然经常被质疑，这两种方法之间的联系从来没有表现出来，本文的目的是提供解决这个问题的新方法。关键的工具是由“Vessiot结构方程“（1903年）的李群或李伪群的转变，其中，相反的“嘉当结构方程“，仍然是未知的今天，并包含“结构常数“，其中一样，在案件中的常数黎曼曲率，一般没有任何关系的任何李代数。主要思想是，然后引入纯差分珍妮特序列$0 \rightarrow Θ\rightarrow T \rightarrow F_0 \rightarrow F_1 \rightarrow ... \rightarrow F_n \rightarrow 0$作为一个决议层$Θ\subset T$的无穷小变换，并诱导一个纯代数“变形序列“与有限维向量空间和线性映射，即使$Θ$是无限维的。几何结构的无穷小等价问题与Janet序列的$ F_0 $处的局部精确性有关，而代数结构的变形问题与$F_1 $的不变部分处的变形序列的精确性有关，即在序列中更进一步，这个意想不到的结果解释了为什么前面引用的许多术语没有成功。最后，我们强调通过例子的一部分，可以发挥计算机代数在任何显式计算。
英文摘要：
In 1953, the physicists E. Inonü and E.P. Wigner introduced the concept of deformation of a Lie algebra by claiming that the limit $1/c \rightarrow 0$, when c is the speed of light, of the composition law $(u,v) \rightarrow (u+v)/(1+(uv/c^2))$ of speeds in special relativity (Poincaré group) should produce the composition law $(u,v) \rightarrow u + v $ used in classical mechanics (Galilée group). However, the dimensionless composition law $(u'=u/c,v'=v/c) \rightarrow (u'+v')/(1+u'v')$ 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 $0 \rightarrow Θ\rightarrow T \rightarrow F_0 \rightarrow F_1 \rightarrow ... \rightarrow F_n \rightarrow 0$ 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.
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