Mathematical Sciences: Leibniz Cohomology, Differential Geometry and Foliations

[704891] 20世纪70年代的一个主要研究课题是叶与李代数上同调之间的相互作用。叶状表示空间上偏微分方程的解，检测给定叶状的上同调类提供了一个简洁的数值不变量来测量方程解集的（全局）扭转。以这种方式发现的第一个上同调类是哥德亿-维不变量，这是三维空间中的一个单一类，它是底层空间的经典de Rham上同调群的一个元素。（对于余维叶变换，没有其他不变量。）这个项目是基于一种全新的计算上同调的方法，它不需要de Rham上同调所需要的经典对称性。这种结构的想法来自法国斯特拉斯堡的让-路易斯·洛迪的工作，被称为莱布尼茨上同论。初步计算的结果是惊人的。仅在余维数为1的情况下，就有无限族的不变量。它们从哥德亿维不变量开始，然后继续到4n和4n+3维度（n任意正整数）。尽管这些新类别是由叶理的单参数变化产生的，但从物理性质的角度解释它们的意义仍然是一个悬而未决的问题。一个更普遍的数学研究课题仍然是根据近150年前黎曼提出的标准对空间进行分类。在过去的一个世纪里取得了很大的进展，但分类并不完全。莱布尼茨上同调作为一种意想不到的、高度非平凡的空间不变量，将成为这一领域的新工具。为了帮助解释莱布尼茨上同，在更熟悉的空间中构造这个不变量将会有所帮助。用来定义莱布尼茨群的形式主义很自然地出现在量子场论中，这是很幸运的。根据V. Kac的工作，场论的元素形成了一个经典的李代数——但只有在一定数量的元素被设为零之后。如果这些元素不被设为零，则原始元素形成的莱布尼茨代数缺乏李代数的对称性。这种莱布尼兹代数的上同调群将成为量子场论的新不变量。＊＊＊

9704891 Lodder A major topic of research in the 1970's was the interplay between foliations and Lie algebra cohomology. A foliation represents the solution of a partial differential equation on a space, and the cohomology classes that detect a given foliation provide a concise numerical invariant measuring the (global) twisting of the solution set of the equation. The first cohomology class to be discovered in this way was the Godbillon-Vey invariant, a single class in dimension three that is an element of the classical de Rham cohomology group of the underlying space. (For codimension one foliations there are no other invariants.) This project is based on a fundamentally new method of computing cohomology that does not require the classical symmetries needed for de Rham cohomology. The ideas for this construction arise from work of Jean-Louis Loday of Strasbourg, France, and are referred to as Leibniz cohomology. The results of preliminary computations are striking. In the codimension one case alone, there are infinite families of invariants for foliations. They begin with the Godbillon-Vey invariant and then continue in dimensions 4n and 4n+3 (n any positive integer). Although the new classes result from a one-parameter variation of a foliation, it remains an open question to interpret their meaning in terms of physical properties. A more general topic of research in mathematics remains to classify space according to the criteria set forth by Riemann nearly 150 years ago. Much progress has been made in the past century, but the classification is not complete. As an unexpected and highly non-trivial invariant of space, Leibniz cohomology will be a new tool in this field. To aid in the interpretation of Leibniz cohomology, a construction for this invariant in terms of more familiar spaces would be helpful. It is auspicious that the formalism used to define the Leibniz groups appears quite naturally in quantum field theory. By work of V. Kac, the elements of a field theory form a classical Lie algebra---but only after a certain number of them are set to zero. If these elements are not set to zero, the original elements form a Leibniz algebra lacking the symmetries of a Lie algebra. The cohomology groups of this Leibniz algebra will be new invariants of quantum field theory. ***