# Lie algebra cohomology

**Lie algebra cohomology** is a cohomology theory for Lie algebras: for a [Lie algebra](https://www.edgepedia.org/lie-algebra) \(\mathfrak{g}\) and a \(\mathfrak{g}\)-module \(V\), it assigns graded vector spaces \(H^{\bullet}(\mathfrak{g},V)\) computed as the cohomology of the Chevalley–Eilenberg complex of alternating multilinear maps on \(\mathfrak{g}\) with values in \(V\)<sup>[1](https://math.mit.edu/~etingof/lnlg.pdf)</sup>. The theory was first introduced in 1929 by [Élie Cartan](https://www.edgepedia.org/elie-cartan) to study the topology of Lie groups and homogeneous spaces, relating the cohomological methods of [Georges de Rham](https://www.edgepedia.org/georges-de-rham) to properties of the Lie algebra, and was extended in 1948 by Claude Chevalley and Samuel Eilenberg to coefficients in an arbitrary Lie module<sup>[2](https://www.mathematik.tu-darmstadt.de/media/algebra/dafra/notizen/2018-11-15_Zorbach_liecohomology.pdf)</sup>.

| Key fact | Detail |
|---|---|
| Standard complex | \(CE^{\bullet}(\mathfrak{g},V) := \mathrm{Hom}(\wedge^{\bullet}\mathfrak{g},V)\), the alternating \(p\)-linear maps \(\mathfrak{g}^p \to V\), with the Chevalley–Eilenberg differential<sup>[1](https://math.mit.edu/~etingof/lnlg.pdf)</sup><sup> • </sup><sup>[3](https://www.numdam.org/item/10.24033/bsmf.2788.pdf)</sup> |
| Degree 0 and 1 | \(H^0(\mathfrak{g},V)=V^{\mathfrak{g}}\), the invariants; \(H^1(\mathfrak{g},M)\) is derivations \(\mathfrak{g}\to M\) modulo inner derivations<sup>[1](https://math.mit.edu/~etingof/lnlg.pdf)</sup><sup> • </sup><sup>[4](https://doc-gitlab.sagemath.org/html/en/reference/categories/sage/categories/finite_dimensional_lie_algebras_with_basis.html)</sup> |
| Degree 2 | \(H^2(\mathfrak{g},M)\) classifies equivalence classes of Lie algebra extensions of \(\mathfrak{g}\) by \(M\)<sup>[4](https://doc-gitlab.sagemath.org/html/en/reference/categories/sage/categories/finite_dimensional_lie_algebras_with_basis.html)</sup> |
| Whitehead's lemma | For semisimple \(\mathfrak{g}\) and finite-dimensional \(V\), \(H^1(\mathfrak{g},V)=H^2(\mathfrak{g},V)=0\)<sup>[1](https://math.mit.edu/~etingof/lnlg.pdf)</sup> |
| Relation to group geometry | For \(\mathfrak{g}\) the Lie algebra of a compact group setting, \(H^{\bullet}(G/K,\mathbb{C}) \cong H^{\bullet}(\mathfrak{g},\mathfrak{t})^{K/K^{\circ}}\), the relative Lie algebra cohomology<sup>[5](https://ocw.mit.edu/courses/18-755-lie-groups-and-lie-algebras-ii-spring-2024/mit18_755_s24_lec_full.pdf)</sup> |
| Computational tool | The Hochschild–Serre spectral sequence has \(E_2^{p,q}=H^p(\mathfrak{G}/\mathfrak{H},H^q(\mathfrak{H},V))\) converging to \(H^*(\mathfrak{G},V)\)<sup>[6](https://encyclopediaofmath.org/wiki/Cohomology_of_Lie_algebras)</sup> |

## The Chevalley–Eilenberg complex

Let \(\mathfrak{g}\) be a Lie algebra and \(V\) a \(\mathfrak{g}\)-module. The Chevalley–Eilenberg (or standard) complex with coefficients in \(V\) is

\[
CE^{\bullet}(\mathfrak{g},V):=\mathrm{Hom}(\wedge^{\bullet}\mathfrak{g},V),
\]

so its \(p\)-cochains are the alternating \(p\)-linear maps \(\mathfrak{g}^p \to V\); in particular \(C^0(\mathfrak{g},V)=V\) and \(C^1(\mathfrak{g},V)\) is the space of linear maps \(\mathfrak{g}\to V\)<sup>[1](https://math.mit.edu/~etingof/lnlg.pdf)</sup><sup> • </sup><sup>[3](https://www.numdam.org/item/10.24033/bsmf.2788.pdf)</sup>. The differential is given by the Cartan formula

\[
d\omega(a_0,\ldots,a_m)=\sum_i (-1)^i a_i\,\omega(a_0,\ldots,\hat a_i,\ldots,a_m)+
\sum_{i<j}(-1)^{i+j}\,\omega([a_i,a_j],a_0,\ldots,\hat a_i,\ldots,\hat a_j,\ldots,a_m),
\]

where the hat marks an omitted argument<sup>[1](https://math.mit.edu/~etingof/lnlg.pdf)</sup>. The cohomology of this complex is the **cohomology of \(\mathfrak{g}\) with coefficients in \(V\)**, written \(H^{\bullet}(\mathfrak{g},V)\); the unadorned notation \(H^{\bullet}(\mathfrak{g})\) means coefficients in the trivial module \(\mathbb{C}\)<sup>[1](https://math.mit.edu/~etingof/lnlg.pdf)</sup>.

The condition \(d^2=0\) is not an accident of the formula: it is equivalent to the Jacobi identity. In the dual chain-complex picture, applying \(d^2\) to a triple of generators gives \([[y_1,y_2],y_3]+[[y_2,y_3],y_1]-[[y_1,y_3],y_2]\), so the differential is well defined precisely because \(\mathfrak{g}\) satisfies Jacobi in degree 3<sup>[7](https://math.mit.edu/juvitop/pastseminars/notes_2021_Fall/Natalie.pdf)</sup>. Dually, the cochain complex \(C^{\bullet}_{\mathrm{Lie}}(\mathfrak{g})\) carries the structure of a commutative differential graded algebra, and the Chevalley–Eilenberg construction gives an equivalence between Lie algebras and semifree differential graded (co)algebras generated in a single degree<sup>[7](https://math.mit.edu/juvitop/pastseminars/notes_2021_Fall/Natalie.pdf)</sup>.

When \(\mathfrak{g}\) is the Lie algebra of a [Lie group](https://www.edgepedia.org/lie-group) \(G\) (or its complexification) and \(V\) is finite dimensional, the complex has a geometric model: \(CE^{\bullet}(\mathfrak{g},V) = (\Omega^{\bullet}(G)\otimes V)^{\mathfrak{g}}\), the \(\mathfrak{g}\)-invariant differential forms on \(G\) with values in \(V\), with the de Rham differential<sup>[1](https://math.mit.edu/~etingof/lnlg.pdf)</sup>. This is the form in which Cartan's original topological motivation appears: replacing arbitrary differential forms by invariant ones, and invariant forms by alternating multilinear forms on the Lie algebra<sup>[8](https://oamonitor.ireland.openaire.eu/national/search/publication?pid=10.1090%2Fs0002-9947-1948-0024908-8)</sup>.

## Low degrees

The first few cohomology groups have direct interpretations<sup>[1](https://math.mit.edu/~etingof/lnlg.pdf)</sup><sup> • </sup><sup>[4](https://doc-gitlab.sagemath.org/html/en/reference/categories/sage/categories/finite_dimensional_lie_algebras_with_basis.html)</sup>:

1. \(H^0(\mathfrak{g},V)=V^{\mathfrak{g}}\), the space of \(\mathfrak{g}\)-invariants in \(V\), that is, the elements \(m\in M\) with \(g m = 0\) for all \(g\in\mathfrak{g}\) in the module action.
2. \(H^1(\mathfrak{g},V)\) is the quotient of the space \(Z^1(\mathfrak{g},V)\) of 1-cocycles, linear maps \(\omega:\mathfrak{g}\to V\) satisfying \(\omega([x,y])=x\omega(y)-y\omega(x)\), by the 1-coboundaries of the form \(\omega(x)=xv\). Equivalently, it is derivations modulo inner derivations.
3. \(H^2(\mathfrak{g},M)\) classifies equivalence classes of Lie algebra extensions of \(\mathfrak{g}\) by \(M\).

Cohomology also computes representation-theoretic data: for representations \(V\) and \(W\) of \(\mathfrak{g}\), \(\mathrm{Ext}^1(V,W)=H^1(\mathfrak{g},\mathrm{Hom}_k(V,W))\)<sup>[1](https://math.mit.edu/~etingof/lnlg.pdf)</sup>.

## Vanishing theorems

Two structural results show when the cohomology disappears. If \(\mathfrak{g}\) is the Lie algebra of a compact Lie group \(G\) and \(V\) is a nontrivial irreducible representation, then \(H^i(\mathfrak{g},V)=0\) for all \(i>0\); in particular this holds for any nontrivial irreducible finite-dimensional representation of a semisimple Lie algebra<sup>[1](https://math.mit.edu/~etingof/lnlg.pdf)</sup>. [Whitehead's lemma](https://www.edgepedia.org/whiteheads-lemma-lie-algebra) states that for semisimple \(\mathfrak{g}\) and any finite-dimensional \(V\), \(H^1(\mathfrak{g},V)=H^2(\mathfrak{g},V)=0\)<sup>[1](https://math.mit.edu/~etingof/lnlg.pdf)</sup>. The cohomology is nontrivial in general when \(\mathfrak{g}\) is not semisimple or \(V\) is infinite dimensional<sup>[1](https://math.mit.edu/~etingof/lnlg.pdf)</sup>.

Concrete computations illustrate the contrast. For the semisimple algebra \(\mathfrak{so}(4)\) over \(\mathbb{Q}\) with trivial coefficients, Sage reports cohomology only in degrees 0, 3, and 6 (dimensions 1, 2, and 1), while the two-dimensional [Heisenberg algebra](https://www.edgepedia.org/heisenberg-algebra) has nonzero cohomology in every degree 0 through 5<sup>[4](https://doc-gitlab.sagemath.org/html/en/reference/categories/sage/categories/finite_dimensional_lie_algebras_with_basis.html)</sup>.

## Relative cohomology and homogeneous spaces

For a pair \((\mathfrak{g},\mathfrak{t})\) of Lie algebras, the **relative Chevalley–Eilenberg complex** \(CE^{\bullet}(\mathfrak{g},\mathfrak{t})\) has cohomology called the **relative Lie algebra cohomology** \(H^{\bullet}(\mathfrak{g},\mathfrak{t})\)<sup>[5](https://ocw.mit.edu/courses/18-755-lie-groups-and-lie-algebras-ii-spring-2024/mit18_755_s24_lec_full.pdf)</sup>. In the compact setting \(CE^{\bullet}(\mathfrak{g},K)=CE^{\bullet}(\mathfrak{g},\mathfrak{t})^{K/K^{\circ}}\), which yields

\[
H^{\bullet}(G/K,\mathbb{C}) \cong H^{\bullet}(\mathfrak{g},\mathfrak{t})^{K/K^{\circ}}
\]

as algebras, so the computation of the cohomology of the homogeneous space \(G/K\) reduces to a purely algebraic problem<sup>[5](https://ocw.mit.edu/courses/18-755-lie-groups-and-lie-algebras-ii-spring-2024/mit18_755_s24_lec_full.pdf)</sup>. A further simplification: if an element \(z\in K\) acts by \(-1\) on \(\mathfrak{g}/\mathfrak{t}\), then the invariant exterior powers vanish in odd degrees, the differential vanishes, and \(H^{\bullet}(G/K,\mathbb{C}) \cong (\Lambda^{\bullet}(\mathfrak{g}/\mathfrak{t})^*)^K\), with cohomology present only in even degrees<sup>[5](https://ocw.mit.edu/courses/18-755-lie-groups-and-lie-algebras-ii-spring-2024/mit18_755_s24_lec_full.pdf)</sup>.

Relative cohomology also connects to group-level cohomology through the van Est isomorphism, which relates the continuous cohomology of a Lie group with the relative Lie algebra cohomology of the corresponding pair<sup>[9](https://www.nzdr.ru/data/media/biblio/kolxoz/M/MA/MAg/Burgos%20Gil%20J.I.%20The%20regulators%20of%20Beilinson%20and%20Borel%20%28draft%2C%20AMS%2C%202001%29%28ISBN%200821826301%29%28107s%29_MAg_.pdf)</sup>.

## Spectral sequences and computation

For a Lie subalgebra \(\mathfrak{H}\) of \(\mathfrak{G}\) such that \(\mathfrak{H}\) and \(\mathfrak{G}/\mathfrak{H}\) are free modules, the Hochschild–[Serre spectral sequence](https://www.edgepedia.org/serre-spectral-sequence) has second page \(E_2^{p,q}=H^p(\mathfrak{G}/\mathfrak{H},H^q(\mathfrak{H},V))\) and converges to \(H^*(\mathfrak{G},V)\); these spectral sequences are used extensively in calculating Lie algebra cohomology<sup>[6](https://encyclopediaofmath.org/wiki/Cohomology_of_Lie_algebras)</sup>. Software implementations exist as well: Sage computes Lie algebra cohomology as the cohomology of the Chevalley–Eilenberg cochain complex, currently for trivial coefficients over the base ring<sup>[4](https://doc-gitlab.sagemath.org/html/en/reference/categories/sage/categories/finite_dimensional_lie_algebras_with_basis.html)</sup>.

## Applications

**Extensions and deformations.** The low-degree groups classify algebraic structures: \(H^2(\mathfrak{g},M)\) classes are equivalence classes of extensions of \(\mathfrak{g}\) by \(M\)<sup>[4](https://doc-gitlab.sagemath.org/html/en/reference/categories/sage/categories/finite_dimensional_lie_algebras_with_basis.html)</sup>, and infinitesimal deformations of a Lie algebra are classified by its second cohomology group<sup>[10](https://www.sciencedirect.com/science/article/abs/pii/S0021869324000577)</sup>. This is the mechanism behind deformation-theoretic classifications of families of Lie algebras, such as the kinematical Lie algebras of physics<sup>[11](https://arxiv.org/abs/1711.06111)</sup>.

**Characteristic classes.** Lecomte's cohomological construction associates characteristic classes to every [Lie algebra extension](https://www.edgepedia.org/lie-algebra-extension), and the classical Chern–Weil construction arises in this context from the Atiyah sequence of a principal bundle<sup>[3](https://www.numdam.org/item/10.24033/bsmf.2788.pdf)</sup>. Building on this, secondary characteristic classes of Lie algebra extensions are defined as classes in \(H^{2p-1}(\mathfrak{g},V)\) obtained from pairs of linear sections of the extension projection; they apply in particular to split extensions, that is, semidirect products of Lie algebras, where the primary characteristic classes vanish<sup>[3](https://www.numdam.org/item/10.24033/bsmf.2788.pdf)</sup>.

**Related cohomology theories.** For a differential graded Lie algebra \(L\) with universal enveloping algebra \(A=UL\), Hochschild cohomology satisfies \(HH^*(A;A)\cong H^*(L,A^a)\), the Lie algebra cohomology with coefficients in the adjoint module<sup>[12](https://link.springer.com/article/10.1007/s40590-025-00814-9)</sup>. The theory also extends to Lie superalgebras, where the Chevalley–Eilenberg complex is the dg vector space \(S\mathfrak{g}[1]\otimes M\) with a differential built from the bracket and the module action<sup>[13](https://scholar.tecnico.ulisboa.pt/api/records/AQNHnyuph3U1wkvhYp9T3hJOQN8aSeNAMiSf/file/9326d0d01fcc3fb3c987dd191fb21f25e27db6fe59a07e7530728ec78c1b0267.pdf)</sup>.

## References

1. [Etingof, P. and colleagues. Lectures on Lie Groups and Lie Algebras, Lecture 47 (Lie algebra cohomology), MIT.](https://math.mit.edu/~etingof/lnlg.pdf)
2. [Lie algebra cohomology (history section), TU Darmstadt seminar notes, 2018.](https://www.mathematik.tu-darmstadt.de/media/algebra/dafra/notizen/2018-11-15_Zorbach_liecohomology.pdf)
3. [Wagner, S. (2019). Secondary characteristic classes of Lie algebra extensions. Bulletin de la Société Mathématique de France 147 (3), 443–453.](https://www.numdam.org/item/10.24033/bsmf.2788.pdf)
4. [Sage 9.4 Reference Manual: Finite Dimensional Lie Algebras With Basis, cohomology().](https://doc-gitlab.sagemath.org/html/en/reference/categories/sage/categories/finite_dimensional_lie_algebras_with_basis.html)
5. [MIT 18.755 Lie Groups and Lie Algebras II, Spring 2024, Lecture 46 (relative Chevalley–Eilenberg complex).](https://ocw.mit.edu/courses/18-755-lie-groups-and-lie-algebras-ii-spring-2024/mit18_755_s24_lec_full.pdf)
6. [Cohomology of Lie algebras, Encyclopedia of Mathematics.](https://encyclopediaofmath.org/wiki/Cohomology_of_Lie_algebras)
7. [The Chevalley–Eilenberg complex for Lie algebras, JUVITOP seminar notes, MIT, 2021.](https://math.mit.edu/juvitop/pastseminars/notes_2021_Fall/Natalie.pdf)
8. [Chevalley, C. and Eilenberg, S. (1948). Cohomology theory of Lie groups and Lie algebras, Transactions of the AMS.](https://oamonitor.ireland.openaire.eu/national/search/publication?pid=10.1090%2Fs0002-9947-1948-0024908-8)
9. [Burgos Gil, J. I. The Regulators of Beilinson and Borel, AMS, 2001 (van Est isomorphism).](https://www.nzdr.ru/data/media/biblio/kolxoz/M/MA/MAg/Burgos%20Gil%20J.I.%20The%20regulators%20of%20Beilinson%20and%20Borel%20%28draft%2C%20AMS%2C%202001%29%28ISBN%200821826301%29%28107s%29_MAg_.pdf)
10. [Cohomologies of difference Lie groups and the van Est theorem, Journal of Algebra, 2024.](https://www.sciencedirect.com/science/article/abs/pii/S0021869324000577)
11. [Barnes, K. et al. Kinematical Lie algebras via deformation theory, arXiv:1711.06111.](https://arxiv.org/abs/1711.06111)
12. [Lie algebra cohomology and the product in the free loop space homology, Boletín de la Sociedad Matemática Mexicana, 2025.](https://link.springer.com/article/10.1007/s40590-025-00814-9)
13. [Lie Algebra Cohomology (Lie superalgebras), Técnico Lisboa lecture notes.](https://scholar.tecnico.ulisboa.pt/api/records/AQNHnyuph3U1wkvhYp9T3hJOQN8aSeNAMiSf/file/9326d0d01fcc3fb3c987dd191fb21f25e27db6fe59a07e7530728ec78c1b0267.pdf)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Advanced algebraic structures › Homological algebra and K-theory › Cohomology of algebras*

*Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026; Oct 15, 2026 · Edited: Sep 30, 2026; Oct 15, 2026 · Last review: Oct 15, 2026*

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