9.3 Lie algebra cohomology

Lie algebra cohomology is a powerful tool for understanding the structure of Lie algebras and their representations. It combines ideas from exterior algebra and homological algebra to provide insights into central extensions, deformations, and invariant forms.

This section explores the Chevalley-Eilenberg complex, which computes Lie algebra cohomology, and relates it to other cohomology theories. We'll see how these concepts apply to physics, geometry, and algebraic topology.

Lie Algebra and Exterior Algebra

Fundamental Algebraic Structures

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• Lie algebra consists of a vector space equipped with a binary operation called the Lie bracket that satisfies bilinearity, alternating identity, and the Jacobi identity
• Exterior algebra constructs an associative algebra from a given vector space where the product of two elements is antisymmetric
• Invariant forms are multilinear forms on a Lie algebra that are invariant under the adjoint action of the Lie algebra on itself
• Cohomology ring represents the collection of cohomology groups of a topological space or algebraic object with a ring structure induced by the cup product

Applications and Properties

• Lie algebras describe the infinitesimal symmetries of differential equations and play a crucial role in physics (quantum mechanics, gauge theories)
• Exterior algebra finds applications in differential geometry, where differential forms are elements of the exterior algebra of the cotangent space
• Invariant forms on a Lie algebra are closely related to the structure of the corresponding Lie group and its homogeneous spaces
• Cohomology ring encodes important topological invariants and provides a way to study the global structure of a space or algebraic object (de Rham cohomology, group cohomology)

Chevalley-Eilenberg and Koszul Complexes

Definitions and Constructions

• Chevalley-Eilenberg complex is a cochain complex associated with a Lie algebra and a module over it, used to define Lie algebra cohomology
• Koszul complex is a chain complex associated with a commutative ring and a sequence of elements, used to study the depth and regularity of ideals
• Cartan's formula expresses the Lie derivative of a differential form in terms of the exterior derivative and interior product, connecting Lie theory with differential geometry

Cohomological Computations

• Chevalley-Eilenberg complex allows the computation of Lie algebra cohomology, which classifies central extensions, abelian extensions, and deformations of Lie algebras
• Koszul complex provides a resolution of a module over a polynomial ring, enabling the computation of Tor functors and Betti numbers
• Cartan's formula facilitates the computation of the cohomology of homogeneous spaces and the study of invariant differential forms on Lie groups (Maurer-Cartan forms, Killing forms)

Relative Cohomology and Whitehead's Theorem

Relative Cohomology Theory

• Relative cohomology extends the notion of cohomology to pairs of spaces or algebras, allowing the study of relative invariants and exact sequences
• Relative Lie algebra cohomology considers cochains that vanish on a given subalgebra, leading to long exact sequences relating absolute and relative cohomology groups
• Relative cohomology arises naturally in the study of fiber bundles, where the cohomology of the total space is related to the cohomology of the base and fiber via the Serre spectral sequence

Whitehead's Theorem and Applications

• Whitehead's theorem for Lie algebras states that a morphism of Lie algebras inducing an isomorphism on cohomology is a quasi-isomorphism, providing a cohomological criterion for equivalence
• Whitehead's theorem has important consequences in deformation theory, allowing the classification of infinitesimal deformations of Lie algebras via cohomology
• Whitehead's theorem is a key tool in the study of homotopy theory of Lie algebras and the formality of differential graded Lie algebras (Quillen's rational homotopy theory)