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

  title={On the history of Lie brackets, crossed modules, and Lie-Rinehart algebras},
  author={Johannes Huebschmann},
  journal={Journal of Geometric Mechanics},
  • J. Huebschmann
  • Published 4 August 2022
  • Mathematics
  • Journal of Geometric Mechanics
This is an overview of ideas related to brackets in early homotopy theory, crossed modules, the obstruction 3-cocycle for the nonabelian extension problem, the Teichmuller cocycle, Lie-Rinehart algebras, Lie algebroids, and differential algebra. 

From Lie algebra crossed modules to tensor hierarchies

A Lie-Rinehart algebra in general relativity

We construct a Lie-Rinehart algebra over an infinitesimal extension of the space of initial value fields for Einstein's equations. The bracket relations in this algebra are precisely those of the



Extensions of Lie Algebras and the Third Cohomology Group

  • S. Goldberg
  • Mathematics
    Canadian Journal of Mathematics
  • 1953
Cohomology theories of various algebraic structures have been investigated by several authors. The most noteworthy are due to Hochschild, MacLane and Eckmann, Chevalley and Eilenberg, who developed

The meaning of the third cocycle in the group cohomology of nonabelian gauge theories

Nonabelian holomorphic Lie algebroid extensions

We classify nonabelian extensions of Lie algebroids in the holomorphic category. Moreover we study a spectral sequence associated to any such extension. This spectral sequence generalizes the

Lectures on Fibre Bundles and Differential Geometry

I Differential calculus.- II Differentiable bundles.- III Connections on principal bundles.- IV Holonomy groups.- V Vector bundles and derivation laws.- VI Holomorphic connections.- References.

Lie-Rinehart algebras, . . .

A Lie-Rinehart algebra (A, L) consists of a commutative algebra A and a Lie algebra L with additional structure which generalizes the mutual structure of interaction between the algebra of smooth

Combinatorial isomorphisms and combinatorial homotopy equivalences

Lie-Rinehart algebras, descent, and quantization

A Lie-Rinehart algebra consists of a commutative algebra and a Lie algebra with additional structure which generalizes the mutual structure of interaction between the algebra of functions and the Lie

Normality of algebras over commutative rings and the Teichmüller class. II.

Using a suitable notion of normal Galois extension of commutative rings, we develop the relative theory of the generalized Teichmüller cocycle map. We interpret the theory in terms of the Deuring

Duality for base-changing morphisms of vector bundles, modules, Lie algebroids and Poisson structures

Abstract The main result of this paper is an extension to Poisson bundles [4] and Lie algebroids of the classical result that a linear map of Lie algebras is a morphism of Lie algebras if and only if

Nonabelian Algebraic Topology: Filtered Spaces, Crossed Complexes, Cubical Homotopy Groupoids

This talk gave a sketch of the contents and background to a book with the title `Nonabelian algebraic topology' being written under support of a Leverhulme Emeritus Fellowship (2002-2004) by the