Lie Algebra CohomologyEdit
Lie algebra cohomology is a mathematical framework that assigns to a Lie algebra and its representations a sequence of abelian groups or vector spaces, capturing structural features such as invariants, extensions, and deformations. Developed in the work of Chevalley and Eilenberg in the 1940s, it provides a precise way to formalize how a symmetry algebra acts on objects and what obstructions arise when one tries to build larger objects from smaller ones. Today the subject sits at the crossroads of algebra, geometry, and mathematical physics, and it plays a central role in understanding how symmetry governs both algebraic structures and geometric or physical models. See for example Lie algebra and cohomology for foundational background, and Chevalley–Eilenberg cohomology as the principal construction behind Lie algebra cohomology.
The central idea is to study the action of a Lie algebra g on a module M (a representation of g) and to organize all possible multilinear, alternating maps from g to M into a single complex whose cohomology measures the failure of certain algebraic equations to have solutions. This viewpoint makes precise the intuition that cohomology classifies obstructions: certain algebraic problems that cannot be solved inside M are detected by nontrivial cohomology classes.
Foundations
A Lie algebra g is a vector space equipped with a bilinear bracket [·,·] that satisfies antisymmetry and the Jacobi identity. The algebra is often considered over a field k of characteristic zero, though many constructions extend to other settings. See Lie algebra for a general introduction and standard examples such as sl(2) or the Heisenberg algebra.
A g-module M (also called a representation) is a vector space equipped with a action of g that is compatible with the Lie bracket. This action is written as x · m for x ∈ g and m ∈ M and satisfies the usual linearity and compatibility conditions. See module (mathematics) and representation theory for related notions.
The Chevalley–Eilenberg cochain complex is built from alternating multilinear maps from g to M. For each n ≥ 0, the n-th cochains are C^n(g,M) = Hom_k(Λ^n g, M), where Λ^n g is the n-th exterior power of g. The differential δ maps C^n(g,M) to C^{n+1}(g,M) and is defined by a standard formula that combines the g-action on M with the Lie bracket on g.
The differential δ satisfies δ^2 = 0, so its kernel modulo its image produces the cohomology groups H^n(g,M) = Ker(δ: C^n → C^{n+1}) / Im(δ: C^{n-1} → C^n). These groups can be described in terms of cocycles and coboundaries: n-cocycles are maps f: Λ^n g → M satisfying a compatibility condition, and n-coboundaries are those arising as δ of an (n−1)-cochain.
In many situations one works with the trivial action on M (the case where g acts trivially on M). The resulting cohomology encodes invariants of the Lie algebra action on scalars and can be interpreted in various geometric or physical contexts.
Low-degree interpretations
H^0(g,M) measures the g-invariant elements of M. Concretely, m ∈ M lies in H^0 if x · m = 0 for all x ∈ g. This is the simplest stable piece of the theory and often identifies fixed points under the symmetry described by g.
H^1(g,M) classifies derivations from g to M up to inner derivations. A derivation is a linear map φ: g → M that satisfies φ([x,y]) = x · φ(y) − y · φ(x). When M = g with the adjoint action, H^1(g,g) encodes outer derivations and, in particular, measures possible infinitesimal automorphisms of g not coming from inner symmetries.
H^2(g,M) has a particularly concrete and important interpretation: it classifies equivalence classes of extensions of Lie algebras of the form 0 → M → E → g → 0 with M regarded as an abelian ideal. In other words, cohomology detects how g can be glued to M to form a larger symmetry algebra, with different classes corresponding to nonisomorphic extension structures.
H^3(g,M) contains higher obstructions to integrating infinitesimal data into genuine algebraic or geometric objects. In deformation theory, H^3 often governs obstructions to extending a first-order deformation to higher orders.
Construction in more detail
The differential δ uses the g-action on M and the Lie bracket on g to combine inputs in a way that mirrors the familiar differential in de Rham cohomology, but in a purely algebraic setting. The explicit formula (in a compact form) has two pieces: the action term that moves one argument at a time by the g-action on M, and the bracket term that encodes the Lie algebra structure through [x_i,x_j]. These two parts ensure δ^2 = 0, making the cochain groups into a cochain complex.
Cochains can be interpreted as alternating multilinear maps, which makes the theory algebraic in flavor, yet the outcomes have geometric and topological consequences when g arises from symmetries of a space or a physical system.
Although presented for finite-dimensional Lie algebras over fields of characteristic zero, many aspects extend to infinite-dimensional, graded, or super Lie algebras, with appropriate adjustments to the cochain spaces and the differential.
Examples and computations
Abelian Lie algebras: If g is abelian and M carries the trivial action, then Λ^n g is just the n-th exterior power of g, and the differential is zero. In this case, H^n(g,M) ≅ Λ^n g^* ⊗ M. This makes the cohomology explicitly computable in simple settings and provides a baseline for comparing with nonabelian cases.
Semisimple Lie algebras: Whitehead’s lemmas state that for a finite-dimensional semisimple g and a finite-dimensional g-module M, H^1(g,M) = 0 and H^2(g,M) = 0 in characteristic zero. This vanishing has strong consequences, for example in the rigidity of semisimple structures: there are no nontrivial infinitesimal deformations or central extensions in these contexts. See Whitehead lemma for a detailed statement and proof.
Central extensions: The second cohomology H^2(g,k) with trivial coefficients classifies central extensions of g by k. Such extensions are exact sequences 0 → k → E → g → 0 where k sits in the center of E. These constructions appear in various places, including the study of physical symmetry algebras and in the realization of algebras as symmetries of higher-dimensional objects. See also central extension.
Heisenberg-like algebras: For certain nonabelian g with a central element, the second cohomology can be nontrivial, reflecting the possibility of nontrivial central extensions even when g is not semisimple. Examples illustrating these phenomena are often discussed in surveys on Lie algebra cohomology and its applications.
Deformation interpretation: When M = g with the adjoint action, H^2(g,g) governs infinitesimal deformations of the Lie bracket. Nontrivial classes correspond to ways to deform the algebra structure to first order, with higher cohomology groups controlling obstructions to extending these deformations to full, non-linear deformations. See deformation theory for related perspectives.
Applications and connections
Extensions and obstructions: Lie algebra cohomology provides a natural home for questions about how a symmetry algebra can be extended by a module or by another algebra, and what obstructions arise to performing these extensions in a coherent way.
Connections to geometry and topology: When a Lie algebra arises from the tangent space at the identity of a Lie group, Lie algebra cohomology relates to de Rham cohomology and to the cohomological properties of the space on which the group acts. The van Est isomorphism, for suitable Lie groups, connects Lie group cohomology with Lie algebra cohomology in low degrees, illustrating the bridge between algebra and topology. See Lie group and group cohomology for broader context.
Representation theory and physics: The cohomology groups capture invariants of representations and classify possible extensions and deformations that can alter the symmetry structure of a model. In physics, symmetry algebras and their central extensions can have physical consequences, such as in quantum-mechanical systems where a central charge appears in the algebra of observables.
Generalizations and related theories: Beyond the classical Lie algebra setting, there are cohomology theories for more sophisticated objects such as Lie superalgebras and L∞-algebras. The latter provide a flexible framework for encoding higher homotopies that appear in deformation theory and modern geometry. Related notions include Hochschild cohomology for associative algebras and various flavors of equivariant or relative cohomology, which connect to broader themes in homological algebra and topology.
Generalizations and further directions
Relative Lie algebra cohomology: Given a Lie algebra g with a subalgebra h, relative cohomology H^*(g,h;M) captures data about cochains that are adapted to the inclusion h ⊂ g and to the h-action on M. This is useful in contexts where one has symmetry breaking or a distinguished subalgebra to compare against.
Lie algebra cohomology with coefficients in different modules: The choice of M affects the nature of the invariants and obstructions. For instance, using the adjoint module g itself highlights deformations of the Lie algebra, while using the trivial module k emphasizes invariants and extensions by scalars.
Lie algebra cohomology and geometry: In differential geometry, Lie algebra cohomology interacts with Cartan's method of moving frames, foliations, and characteristic classes, where symmetry algebras govern geometric structures and their deformations.
Computational approaches: In practice, computing Lie algebra cohomology often relies on exploiting the structure of g (semisimple, nilpotent, abelian) and the representation M. Computer algebra systems are increasingly used to handle complex examples, especially in higher degrees or with infinite-dimensional algebras.