On the history of Lie brackets, crossed modules, and Lie-Rinehart algebras

TL;DR

This paper traces the historical development of Lie brackets, crossed modules, and Lie-Rinehart algebras, highlighting their interconnected evolution across group theory, homotopy, and algebraic structures over time.

Contribution

It provides a comprehensive historical overview of the development and interrelations of Lie brackets, crossed modules, and Lie-Rinehart algebras, emphasizing forgotten and parallel discoveries.

Findings

Historical connections between group theory and homotopy via Whitehead product.

Development of crossed modules in topology and group extensions.

Parallel evolution of Lie algebra concepts and their algebraic structures.

Abstract

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.