9.3 Lie algebra cohomology
3 min read•august 7, 2024
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 .
This section explores the , 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 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
- 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 associated with a Lie algebra and a module over it, used to define Lie algebra cohomology
- is a chain complex associated with a commutative ring and a sequence of elements, used to study the depth and regularity of ideals
- 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
- extends the notion of cohomology to pairs of spaces or algebras, allowing the study of relative invariants and exact sequences
- 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
- 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 , 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)
Key Terms to Review (24)
Borel–Bott Theorem: The Borel–Bott Theorem is a fundamental result in algebraic geometry and topology that connects the cohomology of certain spaces to their geometric properties. Specifically, it relates the cohomology of a complex algebraic variety to the representation theory of its associated Lie algebra, revealing deep connections between algebraic geometry and Lie theory.
Cartan's Formula: Cartan's Formula is a key result in the study of Lie algebra cohomology that relates the cohomology of a Lie algebra to its representation theory. This formula provides a way to compute the cohomology groups of a Lie algebra by using its derived functor, specifically linking the exterior algebra and the Chevalley-Eilenberg complex. It plays an important role in understanding how these algebraic structures interact with differential forms and higher algebraic constructs.
Chern-Weil Theory: Chern-Weil Theory is a mathematical framework that connects differential geometry and topology, specifically relating characteristic classes of vector bundles to curvature forms. It provides a method for computing topological invariants using the geometric data encoded in curvature, allowing us to associate algebraic structures with geometric objects.
Chevalley-Eilenberg Cohomology: Chevalley-Eilenberg cohomology is a powerful tool used in mathematics to study the cohomological properties of Lie algebras. This cohomology theory captures information about the structure and representation of Lie algebras, allowing mathematicians to explore various algebraic and geometric aspects of these mathematical objects. It plays a crucial role in understanding the relationships between Lie algebras and differential forms on manifolds.
Chevalley-Eilenberg Complex: The Chevalley-Eilenberg complex is a construction in homological algebra used to compute the cohomology of a Lie algebra with coefficients in a module. This complex provides a systematic way to derive the cohomology groups of a Lie algebra, connecting the structure of the algebra to its topological and algebraic properties. It plays a crucial role in the study of Lie algebra cohomology, as it allows us to analyze the relationships between different modules associated with the Lie algebra.
Cochain complex: A cochain complex is a sequence of abelian groups (or modules) connected by homomorphisms, which are called coboundary maps, that facilitate the study of cohomology. It is essentially the dual notion to a chain complex and provides a framework to analyze algebraic structures and topological spaces using cohomology theories. By taking the dual of the chain complex, it highlights how cochains can capture information about the structure and properties of spaces in various mathematical contexts.
Cocycles: Cocycles are mathematical objects that arise in the study of cohomology theories, representing elements of a cochain complex that satisfy a certain condition of closure. They are crucial in understanding the relationship between singular cohomology and homology, as well as in group and Lie algebra cohomology. Cocycles can be thought of as the 'good' elements that contribute to the computation of cohomology groups and help in identifying isomorphisms between different algebraic structures.
Cohomology Ring: The cohomology ring is an algebraic structure that combines the concepts of cohomology and ring theory, specifically capturing how topological spaces can be studied through algebraic invariants. It consists of the cohomology groups of a topological space, which are equipped with a ring structure via the cup product, allowing for the combination of cohomology classes. This structure is crucial in various applications across algebra and topology, providing insight into the relationships between different spaces and their properties.
Deformation theory: Deformation theory is a branch of mathematics that studies how mathematical structures, such as algebraic objects or geometric shapes, can be continuously transformed into one another. This concept plays a significant role in understanding the moduli spaces of these structures, allowing mathematicians to analyze and classify various deformations. By examining how small changes can affect the properties of these structures, deformation theory provides insights into their underlying features and symmetries.
Derivation: In mathematics, a derivation is a unary operation on an algebra that generalizes the concept of differentiation. It is a linear map that satisfies the Leibniz rule, which states that the derivation of a product is given by the product rule. This concept plays a critical role in Lie algebra cohomology, where derivations help define the structure and relationships between various algebraic objects.
Differential Graded Algebra: A differential graded algebra is an algebraic structure that combines the concepts of both graded algebra and differential forms. It consists of a graded vector space equipped with a bilinear product that is associative and satisfies the graded Leibniz rule, along with a differential operator that decreases the degree by one and squares to zero. This structure is essential in many areas of mathematics, as it provides a framework for understanding cohomology theories, including those that arise in the study of various algebraic and geometric structures.
Dimension: Dimension refers to a measure of the size or complexity of a mathematical object, often indicating the minimum number of coordinates needed to specify a point within that object. In various contexts, dimension helps in understanding the structural properties of algebraic objects, and it plays a vital role in exploring relationships between them, such as in the classification of homological properties, understanding cohomological features in Lie algebras, and examining the depth and properties of rings.
Extension Problems: Extension problems refer to questions in homological algebra that investigate the conditions under which a given module can be extended by another module. In the context of Lie algebra cohomology, these problems involve examining how modules over Lie algebras can be constructed from simpler modules, emphasizing the relationships between different representations of the algebra and their cohomological properties.
H^n(g): In the context of Lie algebra cohomology, h^n(g) denotes the n-th cohomology group of a Lie algebra g. This concept is crucial as it provides insight into the structure and properties of the Lie algebra, capturing information about its representations and extensions.
Invariant Forms: Invariant forms are mathematical objects that remain unchanged under certain transformations or actions, especially in the context of algebraic structures like Lie algebras. These forms play a crucial role in cohomology theories, where they help to characterize the properties of the algebraic structure being studied, such as understanding its cohomology groups and their relationships with other algebraic entities.
Koszul Complex: The Koszul complex is a specific type of chain complex associated with a sequence of elements in a ring or algebra, often used to study homological properties such as resolutions and cohomology. It captures the relationships between these elements through its structure, enabling the computation of derived functors like Tor and Ext. In contexts involving Lie algebras and homological algebra, the Koszul complex serves as a powerful tool for investigating various algebraic properties and cohomological dimensions.
Lie algebra cohomology: Lie algebra cohomology is a mathematical tool used to study the properties and structures of Lie algebras through the lens of cohomology theory. It allows for the examination of extensions, deformations, and representations of Lie algebras by using derived functors, which help in obtaining new algebraic invariants that provide insight into the algebra's behavior.
Lie bracket: The Lie bracket is an operation defined on a Lie algebra that takes two elements and produces another element of the same algebra, reflecting the algebraic structure of the system. This operation captures the essence of the non-commutative nature of the algebra and is essential for studying the properties and cohomology of Lie algebras. The Lie bracket is bilinear, antisymmetric, and satisfies the Jacobi identity, making it a crucial component in understanding the underlying algebraic frameworks.
Local-to-global principles: Local-to-global principles refer to the concept that local properties of mathematical objects can provide insights into their global behavior or structure. In the context of cohomology, particularly in Lie algebra cohomology, these principles suggest that understanding a structure locally (like at a point) can help in reconstructing or understanding the entire structure globally.
Relative cohomology: Relative cohomology is a type of cohomology that studies the properties of a topological space with respect to a subspace, focusing on the relationships between their structures. This concept allows mathematicians to capture how a space behaves when considered alongside a specific subspace, which is particularly useful in various contexts, including algebraic topology and the study of Lie algebras. It helps in understanding how additional constraints or structures can influence the cohomological properties of the larger space.
Relative Lie Algebra Cohomology: Relative Lie algebra cohomology is a version of cohomology theory applied to a Lie algebra that takes into account a subalgebra and studies how the cohomological properties change when considering the inclusion of this subalgebra. This concept is crucial in understanding the relationship between a Lie algebra and its subalgebras, as it helps reveal how certain algebraic structures can influence or dictate the behavior of cohomology classes. By exploring these relative aspects, one can derive valuable insights into both the algebraic and geometric properties of the Lie algebras involved.
Representation Theory: Representation theory is the mathematical study of how algebraic structures, such as groups and algebras, can be represented through linear transformations of vector spaces. This area of study connects algebra to geometry and analysis, allowing for deeper insights into both the structure of the algebraic entities and their applications in various mathematical contexts, including cohomology and modern research trends in homological algebra.
Triviality: In the context of Lie algebra cohomology, triviality refers to the property of a cohomology class being zero or having no interesting structure. This means that the cohomological features do not provide any new information about the underlying algebraic structure, and hence, the cohomology class can be considered uninteresting. Triviality is essential for understanding the distinctions between non-trivial classes, which reveal deeper insights into the properties of Lie algebras and their representations.
Whitehead's Theorem: Whitehead's Theorem states that for a given Lie algebra, the cohomology of a certain type of algebra is determined by its derived functors. This theorem provides a significant connection between the structure of Lie algebras and their cohomological properties, particularly revealing how the cohomology groups can be computed using specific algebraic methods. This theorem is pivotal in understanding how different representations of Lie algebras can interact with their cohomology.