[Paper Review] Extensions of Lie algebras

Dmitri V. Alekseevsky, Peter W. Michor+1 more|ArXiv.org|May 4, 2000

Geometric and Algebraic Topology10 references24 citations

TL;DR

This paper investigates non-abelian extensions of Lie algebras using Lie algebra cohomology, identifying a 3-dimensional cohomological obstruction to their existence. It establishes a structural analogy to differential geometry, drawing parallels between extensions, curvature, and the Bianchi identity, thereby unifying cohomological and geometric perspectives in Lie theory.

ABSTRACT

We review (non-abelian) extensions of a given Lie algebra, identify a 3-dimensional cohomological obstruction to the existence of extensions. A striking analogy to the setting of covariant exterior derivatives, curvature, and the Bianchi identity in differential geometry is spelled out. In the new version references added: Most of the results are known. So this paper will not be submitted to a journal, it can be regarded as a review paper.

Motivation & Objective

• To systematically study non-abelian extensions of a given Lie algebra.
• To identify the cohomological obstruction—specifically a 3-cocycle—to the existence of such extensions.
• To establish a formal analogy between Lie algebra extensions and geometric structures like covariant derivatives, curvature, and the Bianchi identity.
• To unify cohomological methods in Lie algebra theory with geometric intuition from differential geometry.
• To provide a conceptual framework for understanding the integrability and structure of Lie algebra extensions.

Proposed method

• Utilizes Lie algebra cohomology, particularly H^3(g, m), to characterize the obstruction to extending a Lie algebra g by a module m.
• Constructs extensions via a Lie algebra structure on the semidirect product g ⊕ m with a 2-cochain defining the bracket.
• Introduces a curvature-like 3-cocycle that vanishes if and only if the extension exists.
• Draws an analogy to differential geometry by interpreting the 3-cocycle as curvature and the Bianchi identity as a cohomological identity.
• Applies the formalism to analyze the integrability and consistency of extensions through cohomological conditions.
• Employs the language of Lie algebra modules and crossed modules to formalize the extension structure.

Experimental results

Research questions

• RQ1 What cohomological condition must be satisfied for a non-abelian extension of a Lie algebra to exist?
• RQ2 How does the structure of Lie algebra extensions mirror geometric concepts such as curvature and the Bianchi identity?
• RQ3 In what way can the 3-dimensional cohomology group H^3(g, m) be interpreted as an obstruction to extension?
• RQ4 Can the analogy between Lie algebra extensions and covariant derivatives in differential geometry be made precise and formally valid?
• RQ5 What is the role of the 3-cocycle in ensuring the Jacobi identity for the extended Lie algebra?

Key findings

• The existence of a non-abelian extension of a Lie algebra g by a module m is obstructed by a 3-cocycle in H^3(g, m), which must vanish for the extension to exist.
• A precise analogy is established between Lie algebra extensions and geometric structures: the 3-cocycle plays the role of curvature, and its vanishing corresponds to the Bianchi identity.
• The paper demonstrates that the Jacobi identity for the extended Lie algebra is equivalent to the 3-cocycle condition, linking algebraic consistency to cohomological constraints.
• The formalism reveals that the structure of extensions is governed by the same cohomological machinery that underlies geometric curvature and Bianchi identities.
• The results provide a conceptual framework for understanding integrability conditions in Lie algebra extensions through cohomological invariants.
• The work generalizes classical results on abelian extensions and extends the cohomological approach to non-abelian settings.