Chevalley–Eilenberg complex

Chevalley–Eilenberg complex
Name	Chevalley–Eilenberg complex
Field	Algebraic topology, Representation theory, Homological algebra
Introduced	1948
Authors	Claude Chevalley, Samuel Eilenberg

Chevalley–Eilenberg complex is a cochain complex associated to a Lie algebra that computes Lie algebra cohomology, used across Élie Cartan, Jean Leray, Hermann Weyl, Claude Chevalley, Samuel Eilenberg, and connections to Henri Cartan's work on characteristic classes and to developments in André Weil's algebraic topology. It forms a bridge between the representation theory of Élie Cartan-style Lie algebras, the homological methods of Samuel Eilenberg and Saunders Mac Lane, and applications in deformation problems studied by Murray Gerstenhaber, Dennis Sullivan, and Alexander Grothendieck.

Definition

The Chevalley–Eilenberg complex is defined for a Lie algebra g over a field K and a g-module M; it generalizes constructions appearing in the work of Élie Cartan and the homological algebra of Samuel Eilenberg and Saunders Mac Lane. Given g and M one forms the graded vector space C^*(g;M)=Hom_K(Λ^* g, M) with cochains built from alternating multilinear maps, closely related to ideas in Hermann Weyl's representation theory, Claude Chevalley's structural theory of algebraic groups, and the cohomological frameworks of Jean-Pierre Serre and Armand Borel.

Construction and differential

Construction uses the exterior algebra Λ^* g and the coadjoint actions studied in the context of Élie Cartan and Hermann Weyl. The differential d: C^n(g;M)→C^{n+1}(g;M) is an alternating sum combining the Lie bracket on g (as in Sophus Lie's legacy) and the module action of g on M, echoing formalism in Samuel Eilenberg and Saunders Mac Lane's homological definitions. Explicitly, for f∈Hom_K(Λ^n g,M) the differential includes terms summing over indices with signs determined by permutations, reflecting combinatorial techniques developed by George Pólya and formal exterior calculus considered by Élie Cartan; the operator squares to zero by the Jacobi identity attributed to Sophus Lie and structural properties analyzed in Claude Chevalley's work on Lie algebras and Hermann Weyl's invariant theory.

Cohomology of Lie algebras

The cohomology H^*(g;M) of this complex captures extensions, derivations, and obstructions studied in deformation contexts by Murray Gerstenhaber and later by Maxim Kontsevich. Low-degree groups H^0, H^1, H^2 correspond classically to invariants, derivations modulo inner derivations, and equivalence classes of extensions, relating to classification problems investigated by Évariste Galois-inspired algebraists and later by Nathan Jacobson and Claude Chevalley. Connections to group cohomology studied by Hyman Bass and John Milnor appear via Lie algebra / group correspondences in the spirit of work by André Weil and Alexander Grothendieck on algebraic structures.

Examples and computations

Standard computations include semisimple Lie algebras g over fields of characteristic zero, where Whitehead lemmas of J. H. C. Whitehead yield vanishing results for H^1 and H^2, reflecting representation-theoretic input from Élie Cartan, Hermann Weyl, and classification results of Élie Cartan and Claude Chevalley. For abelian Lie algebras the complex reduces to the de Rham–type exterior algebra exploited in Élie Cartan's differential forms and computations reminiscent of Jean Leray's spectral sequences; for nilpotent Lie algebras explicit bases and cochain models appear in the study of Marius Sophus Lie-related structures and applications in geometry by S. S. Chern and Shing-Tung Yau. Examples computed in low dimensions connect to invariants used in the theory of William Thurston and William Fulton's algebraic geometry work. Koszul duality perspectives related to Jean-Louis Koszul and homotopical algebra frameworks of Daniel Quillen and Dennis Sullivan provide computational tools and comparisons with Hochschild cohomology studied by Gerald Hochschild.

Properties and functoriality

The Chevalley–Eilenberg construction is functorial for Lie algebra homomorphisms, intertwining module pullbacks and pushforwards much as natural transformations appear in the categorical settings developed by Saunders Mac Lane and Max Kelly. It admits spectral sequences analogous to the Hochschild–Serre spectral sequence from work by Gerald Hochschild and Jean-Pierre Serre, and behaves well under direct sums, semidirect products, and universal enveloping algebra relationships studied by Maurice Auslander and I. N. Herstein. Dualities and universal coefficient theorems relate it to de Rham and group cohomology settings explored by Jean Leray and André Weil.

Applications (deformation theory, characteristic classes)

Applications include deformation theory of associative and Lie algebras as in foundational work by Murray Gerstenhaber and later extensions by Maxim Kontsevich; Chevalley–Eilenberg cohomology controls formal deformations, obstruction classes, and moduli problems central to Alexander Grothendieck's ideas. In differential geometry and topology it underlies constructions of characteristic classes in the spirit of Henri Cartan, Shiing-Shen Chern and Raoul Bott, informing index theorems connected to the work of Atiyah–Singer collaborators and to modern developments by Edward Witten and Graeme Segal. The complex also appears in BRST quantization in theoretical physics influenced by Richard Feynman-era quantum field theory and in modern homotopical algebra formulations pursued by Jacob Lurie and Dennis Sullivan.

Category:Lie algebra cohomology