论文标题
关于谎言括号，交叉模块和Lie-rinehart代数的历史
On the history of Lie brackets, crossed modules, and Lie-Rinehart algebras
论文作者
论文摘要
这里的目的是勾勒出与括号和类似概念相关的思想的发展：由于博士学位而导致的一些纯粹的群体理论组合学，导致了同型理论中怀特海产品的雅各比身份的证明。 Whitehead引入了交叉模块，以表征第二个相对同拷贝组。在组合群体理论考虑的指导下，Reidemeister和Peiffer探索了这种结构，以形成正常形式，以分解3个manifold。但是，交叉的模块也潜伏在被遗忘的小组扩展问题的方法后面：图灵稍后由埃伦贝格·马克（Eilenberg-Mac）隔离障碍物3循环，已经证明了eilenberg-mac Lane定理，以使该循环的阶层消失，以使其对解决方案的效果消失，以使其对对应的解决方案的存在相等。这种包裹的旋转旋转与所谓的Teichmueller Cocycle有关。谎言代数有一个平行的发展，其中包括戈德堡的一张遗忘的纸，同样是lie-rinehart代数和谎言代数。在这种情况下，Turing定理的版本多次被发现，几乎没有暗示相互关系。同样，至少在差分代数的文献中至少隐式地存在了长期存在的lie-rinehart代数。
The aim here is to sketch the development of ideas related to brackets and similar concepts: Some purely group theoretical combinatorics due to Ph. Hall led to a proof of the Jacobi identity for the Whitehead product in homotopy theory. Whitehead introduced crossed modules to characterize a second relative homotopy group; guided by combinatorial group theory considerations, Reidemeister and Peiffer explored this kind of structure to develop normal forms for the decomposition of a 3-manifold; but crossed modules are also lurking behind a forgotten approach of Turing to the extension problem for groups: Turing concocted the obstruction 3-cocycle isolated later by Eilenberg-Mac Lane and already proved the Eilenberg-Mac Lane theorem to the effect that the vanishing of the class of that cocycle is equivalent to the existence of a solution for the corresponding extension problem. This Turing cocycle is related to what has come to be known as Teichmueller cocycle. There was a parallel development for Lie algebras including a forgotten paper by Goldberg and, likewise, for Lie-Rinehart algebras and Lie algebroids. Versions of Turing's theorem were discovered several times under such circumstances, and there is rarely a hint at the mutual relationship. Also, Lie-Rinehart algebras have for long occurred in the literature on differential algebra, at least implicitly.
