03.04.22 · modern-geometry / differential-forms

Lie algebroid cohomology and the Chevalley-Eilenberg differential

shipped3 tiersLean: none

Anchor (Master): Mackenzie — General Theory of Lie Groupoids and Lie Algebroids Ch. III §3; Chevalley-Eilenberg — Cohomology theory of Lie groups and Lie algebras (Trans. AMS 63, 1948); Crainic-Fernandes — Integrability of Lie brackets (Ann. Math. 157, 2003)

Intuition Beginner

A Lie algebroid is an infinitesimal system of arrows over a space: at each point you are told which directions you may move, and how those directions combine through a bracket. This unit builds one machine that takes such a system and produces a sequence of cohomology groups — a ledger of which forms are closed but not boundaries. The surprise is how many familiar ledgers turn out to be special cases of this single one.

Feed in the system where every tangent direction is allowed, and the machine prints ordinary de Rham cohomology, the one that counts holes in a smooth space. Feed in a single Lie algebra sitting over one point, and the same machine prints Lie-algebra cohomology, the algebraic ledger Chevalley and Eilenberg wrote down in 1948. Feed in the directions tangent to a Poisson structure, and out comes Poisson cohomology.

So the slogan is: one differential complex, many faces. The form of the answer is fixed once and for all; only the input algebroid changes. Learning the machine once means learning de Rham, Lie-algebra, and Poisson cohomology at the same time, and seeing why they were always the same idea wearing different clothes.

Visual Beginner

Picture a single black box with a slot on the left and a printer on the right. Into the slot you push a card naming which algebroid you are using; from the printer comes a ladder of cohomology groups, one rung per degree.

The three cards on the left all fit the same slot, and the three ladders on the right are built from the same rung-shape. That sameness of shape is the whole point of the picture: the printer never changes, only the card you insert. The box is the algebroid differential; the cards are the choices of algebroid; the ladders are the cohomologies that result.

Worked example Beginner

Take the simplest input: a space where every tangent direction is allowed, with the rule that a direction is its own shadow. This is the tangent algebroid. The forms you act on are ordinary differential forms, and the differential the machine builds is the ordinary one from de Rham's theory. So the output ladder is plain de Rham cohomology, counting the holes of the space.

Now swap the card. Put in a single Lie algebra over one point, with no room to move along any base direction. The forms become alternating multilinear gadgets on the Lie algebra, and the machine's differential is built only from the bracket, since there are no base directions left to use. The ladder that prints is Lie-algebra cohomology, exactly the one Chevalley and Eilenberg defined.

Swap once more. Insert the directions coming from a Poisson bracket. The differential now uses that bracket to push forms up one degree, and the ladder is Poisson cohomology, whose lowest rungs record the symmetries and the deformations of the Poisson structure.

What this shows: three classical theories, one construction. We did not redo any work between the cards. We changed the input, turned the same crank, and read three different ledgers off the same printer.

Check your understanding Beginner

Formal definition Intermediate+

Let $(A, ρ, [\cdot, \cdot])$ be a Lie algebroid 03.04.16 over $M$ , with anchor $ρ : A \to T M$ a bundle map and a Lie bracket on $Γ (A)$ satisfying the Leibniz rule $[X, f Y] = f [X, Y] + (ρ (X) f) Y$ for $X, Y \in Γ (A)$ and $f \in C^{\infty} (M)$ . Write $A^{*}$ for the dual bundle and $Ω^{p} (A) := Γ (Λ^{p} A^{*})$ for the algebroid $p$ -forms, the $C^{\infty} (M)$ -multilinear alternating maps $Γ (A)^{p} \to C^{\infty} (M)$ . Set $Ω^{0} (A) = C^{\infty} (M)$ .

The Lie algebroid de Rham (Chevalley-Eilenberg) differential is the degree- $+ 1$ operator $d_{A} : Ω^{p} (A) \to Ω^{p + 1} (A)$ defined by the Koszul formula $$ \begin{aligned} (d_A\omega)(X_0,\ldots,X_p) &= \sum_{i=0}^{p} (-1)^i, \rho(X_i),\omega(X_0,\ldots,\widehat{X_i},\ldots,X_p) \ &\quad + \sum_{i<j} (-1)^{i+j}, \omega\big([X_i,X_j],,X_0,\ldots,\widehat{X_i},\ldots,\widehat{X_j},\ldots,X_p\big), \end{aligned} $$ where the anchor $ρ (X_{i})$ acts on the function $ω (\dots)$ as a vector field and a hat marks an omitted argument. On functions $f \in Ω^{0} (A)$ this reads $(d_{A} f) (X) = ρ (X) f$ , so $d_{A} f$ is $ρ^{*} df$ .

One checks $d_{A}$ is $R$ -linear and $C^{\infty} (M)$ -multilinear in the form-slot output: the candidate $d_{A} ω$ is genuinely tensorial, because the Leibniz failures from differentiating coefficients in the first sum cancel against the bracket's Leibniz terms in the second sum. The pair $(Ω^{∙} (A), d_{A})$ is the Lie algebroid de Rham complex, and its cohomology $$ H^\bullet(A) := \frac{\ker\big(d_A : \Omega^\bullet(A) \to \Omega^{\bullet+1}(A)\big)}{\operatorname{im}\big(d_A : \Omega^{\bullet-1}(A) \to \Omega^\bullet(A)\big)} $$ is the Lie algebroid cohomology of $A$ (with constant coefficients $R$ ).

Cohomology with coefficients. A representation of $A$ on a vector bundle $E \to M$ is a flat $A$ -connection, an operator $\nabla : Γ (A) \times Γ (E) \to Γ (E)$ with $\nabla_{f X} e = f \nabla_{X} e$ , $\nabla_{X} (f e) = f \nabla_{X} e + (ρ (X) f) e$ , and curvature $R (X, Y) = [\nabla_{X}, \nabla_{Y}] - \nabla_{[X, Y]} = 0$ . Such a $\nabla$ extends $d_{A}$ to a differential $d_{A}^{\nabla}$ on $Ω^{∙} (A; E) = Γ (Λ^{∙} A^{*} \otimes E)$ with $d_{A}^{\nabla}^{2} = 0$ exactly because $R = 0$ , giving $H^{∙} (A; E)$ . The construction of representations is the subject of 03.04.23; here $E$ enters only as the coefficient bundle.

Counterexamples to common slips

The formula is not the ordinary exterior derivative wearing a disguise: when the anchor is not injective, $d_{A}$ sees bracket data invisible to $d$ , and $H^{∙} (A)$ can differ wildly from the base de Rham cohomology.
$d_{A}^{2} = 0$ is not automatic from the bundle structure alone: it is equivalent to the Jacobi identity together with the anchor-Leibniz compatibility, and it fails for an "almost Lie algebroid" whose bracket violates Jacobi.
The cohomology $H^{∙} (A)$ is not always finite-dimensional even over a compact base, unlike de Rham cohomology; Poisson cohomology in the cotangent-algebroid case is the standard source of infinite-dimensional examples.

Key theorem with proof Intermediate+

Theorem ( $d_{A}^{2} = 0$ is the Jacobi identity). Let $A \to M$ be a vector bundle with an anchor $ρ : A \to T M$ and a bracket $[\cdot, \cdot]$ on $Γ (A)$ satisfying the Leibniz rule, so that $ρ$ is a bracket homomorphism. Define $d_{A}$ by the Koszul formula. Then $d_{A}$ is well-defined ( $C^{\infty}$ -tensorial) and $d_{A} \circ d_{A} = 0$ if and only if the bracket satisfies the Jacobi identity. Hence a Lie algebroid carries a genuine differential complex, and conversely a degree-one square-zero $d_{A}$ on $\Gamma(\Lambda^\bullet A^) $e n co d es a L i e a l g e b r o i d s t r u c t u r eo n$ A$.*

Proof. Tensoriality first. For a $1$ -form $ω$ and $X_{0}, X_{1} \in Γ (A)$ , replace $X_{1}$ by $f X_{1}$ . The first sum contributes $ρ (X_{0}) (f ω (X_{1})) - ρ (f X_{1}) ω (X_{0}) = f [ρ (X_{0}) ω (X_{1}) - ρ (X_{1}) ω (X_{0})] + (ρ (X_{0}) f) ω (X_{1})$ , and the bracket term contributes $- ω ([X_{0}, f X_{1}]) = - f ω ([X_{0}, X_{1}]) - (ρ (X_{0}) f) ω (X_{1})$ by the Leibniz rule. The two $(ρ (X_{0}) f) ω (X_{1})$ pieces cancel, leaving $f \cdot (d_{A} ω) (X_{0}, X_{1})$ . The same cancellation, between the anchor-Leibniz term and the bracket-Leibniz term, runs in every degree, so $d_{A} ω$ is tensorial.

Now compute $d_{A}^{2}$ on a function $f \in Ω^{0} (A)$ . We have $(d_{A} f) (X) = ρ (X) f$ , so $$ (d_A^2 f)(X,Y) = \rho(X)\rho(Y)f - \rho(Y)\rho(X)f - \rho([X,Y])f = \big([\rho(X),\rho(Y)] - \rho([X,Y])\big)f. $$ This vanishes for all $f$ exactly when $ρ$ is a bracket homomorphism, which the Leibniz rule already forces. So $d_{A}^{2} = 0$ in degree $0$ is the homomorphism property of the anchor.

In degree $1$ , evaluate $(d_{A}^{2} ω) (X, Y, Z)$ for $ω \in Ω^{1} (A)$ . Expanding $d_{A} (d_{A} ω)$ and collecting terms, the anchor-anchor pieces reorganise into commutators $[ρ (X), ρ (Y)]$ which, by the just-proven homomorphism property, become $ρ ([X, Y])$ ; these combine with the anchor-on-bracket pieces to cancel in pairs. The residue is precisely $$ (d_A^2\omega)(X,Y,Z) = -,\omega\big([[X,Y],Z] + [[Y,Z],X] + [[Z,X],Y]\big), $$ the cyclic Jacobiator paired against $ω$ . Therefore $d_{A}^{2} ω = 0$ for every $1$ -form $ω$ if and only if the Jacobi identity holds. Because $d_{A}$ is a degree-one derivation generated by its action on $Ω^{0}$ and $Ω^{1}$ — it satisfies $d_{A} (α \land β) = d_{A} α \land β + (- 1)^{∣ α ∣} α \land d_{A} β$ , checked directly from the Koszul formula — vanishing of $d_{A}^{2}$ on degrees $0$ and $1$ propagates to all degrees. So $d_{A}^{2} = 0$ across the whole complex is equivalent to Jacobi.

For the converse, given a square-zero degree-one derivation $D$ on $Γ (Λ^{∙} A^{*})$ , define $ρ$ by $ρ (X) f := (D f) (X)$ and recover the bracket from $D$ on $1$ -forms; tensoriality of $D$ forces the Leibniz rule and $D^{2} = 0$ forces Jacobi, reconstructing the algebroid. $□$

Bridge. This theorem is exactly the statement that the algebroid axioms and the differential are two faces of one object, and it builds toward the integration story by furnishing the cohomology in which obstructions live: the foundational reason a Lie algebroid has a cohomology theory at all is that Jacobi is precisely $d_{A}^{2} = 0$ . The construction generalises both the ordinary exterior derivative — recovered when $A = T M$ , where the Jacobi identity for vector fields is the classical $d^{2} = 0$ — and the Chevalley-Eilenberg differential of a Lie algebra, recovered over a point. Putting these together, the central insight is that closedness and exactness for $d_{A}$ measure, leaf by leaf and degree by degree, the same data that controls whether $A$ integrates; this is dual to the bracket picture, and it appears again in the integrability obstruction $H^{2} (A; ker ρ)$ of 03.04.18.

Exercises Intermediate+

Exercise 5 (medium, proof).

Prove that $d_{A}$ is a degree-one derivation: $d_{A} (α \land β) = d_{A} α \land β + (- 1)^{∣ α ∣} α \land d_{A} β$ for $α \in Ω^{p} (A)$ , $β \in Ω^{q} (A)$ .

Hint

It suffices to check on generators $f \in Ω^{0}$ and $ω \in Ω^{1}$ , since wedge products of these span.

Answer

Both sides are $R$ -bilinear, so check on generators. On $Ω^{0} \times Ω^{0}$ : $d_{A} (f g) (X) = ρ (X) (f g) = (ρ (X) f) g + f (ρ (X) g) = (d_{A} f g + f d_{A} g) (X)$ , using that $ρ (X)$ is a derivation. On $Ω^{0} \times Ω^{1}$ : a direct expansion of $(d_{A} (f ω)) (X, Y)$ against $(d_{A} f \land ω + f d_{A} ω) (X, Y)$ matches term by term, again because $ρ (X)$ differentiates the coefficient $f$ . Since every form is a $C^{\infty} (M)$ -combination of wedge powers of $1$ -forms, and the Leibniz rule is multiplicative, the derivation identity extends to all degrees with the sign $(- 1)^{∣ α ∣}$ tracking the passage of $d_{A}$ past $α$ .

Exercise 6 (hard, proof).

Let $A$ be a regular Lie algebroid with injective anchor onto an integrable distribution $D = im ρ \subseteq T M$ (a foliation $F$ ). Show that $H^{∙} (A)$ is the foliated (leafwise) de Rham cohomology $H^{∙} (F)$ .

Hint

An injective anchor identifies $A$ with $D = T F$ as a Lie algebroid; forms on $A$ are leafwise forms.

Answer

Injectivity of $ρ$ makes $ρ : A \sim D$ a bundle isomorphism onto the involutive distribution $D = T F$ , and it intertwines the brackets because $ρ$ is a bracket homomorphism; so $A ≅ T F$ as Lie algebroids. Then $Ω^{∙} (A) ≅ Γ (Λ^{∙} (T F)^{*})$ is the space of leafwise forms, and $d_{A}$ corresponds under this isomorphism to the leafwise exterior derivative $d_{F}$ , which differentiates only along the leaves. Its cohomology is by definition the foliated de Rham cohomology $H^{∙} (F)$ . When $F$ has a single leaf $M$ this collapses to ordinary $H_{dR}^{∙} (M)$ , consistent with the $T M$ case.

Exercise 7 (hard, proof).

Show that $H^{1} (A)$ classifies the outer derivations of the algebroid that act as functions: precisely, $H^{1} (A) = {X \in Γ (A) : ρ (X) = 0, [X, \cdot] = 0}^{*}$ -style data fails, so instead prove that a class in $H^{1} (A)$ is a derivation of $C^{\infty} (M)$ along $ρ$ modulo inner ones, and interpret $H^{0} (A)$ as the $ρ$ -invariant functions.

Hint

$H^{0} (A) = ker (d_{A} : Ω^{0} \to Ω^{1})$ ; a closed $1$ -form modulo exact ones is the relevant quotient for $H^{1}$ .

Answer

$H^{0} (A) = {f : d_{A} f = 0} = {f : ρ (X) f = 0 \forall X}$ , the functions constant along the image distribution of the anchor — the $ρ$ -invariant (Casimir-type) functions. For $H^{1}$ , a closed $1$ -form $ω$ satisfies $ρ (X) ω (Y) - ρ (Y) ω (X) = ω ([X, Y])$ , which says $ω$ is a $C^{\infty} (M)$ -linear cocycle measuring how a chosen "potential" fails to be exact; exact ones are $ω = d_{A} f$ . The quotient $H^{1} (A) = Z^{1} / B^{1}$ thus records the closed-but-not-exact potentials, equivalently the derivations $X \mapsto ω (X)$ of the function algebra compatible with $ρ$ modulo those of the form $X \mapsto ρ (X) f$ , i.e. outer derivations modulo inner. Over a point this is the abelianisation pairing $H^{1} (g) = (g_{ab})^{*}$ of Exercise 3.

Lean formalization Master

lean_status: none — Mathlib has ExteriorAlgebra, the Lie-algebra Chevalley-Eilenberg complex LieAlgebra.Cohomology, and a developing vector-bundle / TangentBundle library, but no LieAlgebroid structure and hence no fibred differential $d_{A}$ on $Γ (Λ^{∙} A^{*})$ . The sketch below names the missing pieces; it does not compile against current Mathlib, which is why no Lean module is declared.

-- Pseudo-Lean: the algebroid de Rham complex, not in Mathlib.
variable {M : Type*} [Manifold M]
variable (A : LieAlgebroid M)          -- anchor ρ : A → TM, bracket on Γ(A)

def AForm (A : LieAlgebroid M) (p : ℕ) := Section (exteriorPower p (dual A))

def d_A (A) {p} (ω : AForm A p) : AForm A (p+1) :=
  fun X => (∑ i, (-1)^i • A.anchor (X i) • ω (omit i X))
         + (∑ i < j, (-1)^(i+j) • ω (cons (A.bracket (X i) (X j)) (omit i j X)))

theorem d_A_sq_zero (A) : ∀ {p} (ω : AForm A p), d_A A (d_A A ω) = 0
  -- equivalent to A.jacobi ∧ A.leibniz; the mathematical content of the unit

def algebroidCohomology (A) (p : ℕ) :=
  (ker (d_A A : AForm A p → AForm A (p+1))) ⧸ (range (d_A A : AForm A (p-1) → AForm A p))

The first genuine obstacle is AForm with the $C^{\infty} (M)$ -module structure on sections of $Λ^{p} A^{*}$ ; the second is d_A_sq_zero, whose proof is the Jacobi/Leibniz computation of the Key theorem. Coefficient cohomology algebroidCohomology A E requires the flat $A$ -connection of 03.04.23. None of these objects exists in Mathlib.

Advanced results Master

The Poisson / Lichnerowicz case. Let $(M, π)$ be a Poisson manifold, with $π \in Γ (Λ^{2} T M)$ the Poisson bivector. Its cotangent algebroid 03.04.19 is $A = T^{*} M$ with anchor $ρ = π^{♯} : T^{*} M \to T M$ and the Koszul bracket on $1$ -forms. Here algebroid forms are multivector fields, $Ω^{p} (A) = Γ (Λ^{p} T M)$ , and the differential $d_{A}$ is $- [π, \cdot]$ , the Schouten bracket with $π$ . Its cohomology is Poisson (Lichnerowicz) cohomology $H_{π}^{∙} (M)$ : $H_{π}^{0}$ is the Casimir functions, $H_{π}^{1}$ is the Poisson vector fields modulo Hamiltonian ones, and $H_{π}^{2}$ controls infinitesimal deformations of $π$ ^{[Lichnerowicz 1977]}. These groups are typically infinite-dimensional, the cleanest illustration that algebroid cohomology is strictly richer than de Rham.

The integrability obstruction lives in $H^{2}$ . For a transitive Lie algebroid the adjoint sequence $0 \to ker ρ \to A ρ T M \to 0$ gives a coefficient complex with values in the isotropy bundle $ker ρ$ . The class obstructing a global splitting compatible with the bracket — and, by the integration story of 03.04.18, the obstruction packaged into the monodromy — is represented by the curvature of a connection on the sequence and lives in $H^{2} (A; ker ρ)$ , the second algebroid cohomology with coefficients in the isotropy ^{[Mackenzie 1987]}. The Chevalley-Eilenberg complex of this unit is precisely the home that 03.04.18 named but did not construct: the curvature $2$ -form there is a $d_{A}$ -cocycle here, and changing the connection changes it by a $d_{A}$ -coboundary, so its class is the well-defined obstruction.

The van Est map and integration. When $A = A (G)$ is the algebroid of a Lie groupoid $G$ , a van Est map relates the differentiable groupoid cohomology $H_{diff}^{∙} (G)$ to the algebroid cohomology $H^{∙} (A)$ , an isomorphism in low degrees when the source fibres are cohomologically simple ^{[Crainic-Fernandes 2003]}. This is the algebroid analogue of the van Est isomorphism between Lie-group and Lie-algebra cohomology, and it threads the cohomology theory directly into the Crainic-Fernandes integration machinery: the integration obstruction is a groupoid-cohomology class pulled back to $H^{2}$ of the algebroid.

The modular class (pointer). Every Lie algebroid $A$ carries a canonical class $mod (A) \in H^{1} (A)$ , the modular class, measuring the failure of $A$ to preserve a volume on $Λ^{top} A \otimes Λ^{top} T^{*} M$ . For the cotangent algebroid it reduces to the modular vector field of a Poisson manifold, and for $A = T M$ it vanishes. It is the first secondary characteristic class detected by $d_{A}$ , and we record it here as the natural next invariant rather than developing it.

Synthesis. A single Koszul formula, with $d_{A}^{2} = 0$ equivalent to the Jacobi identity, is the foundational reason a Lie algebroid carries a cohomology theory, and putting the examples together shows that de Rham, Chevalley-Eilenberg, Poisson-Lichnerowicz, and foliated cohomology are one construction read off four input algebroids. This is exactly the unification the beginner slogan promised: $A = T M$ gives de Rham 03.04.06, $A = g$ gives Lie-algebra cohomology 03.04.01, the cotangent algebroid gives Poisson cohomology, and an injective anchor gives foliated cohomology. The central insight is that the second cohomology with isotropy coefficients, $H^{2} (A; ker ρ)$ , is the home of the integration obstruction of 03.04.18 — the curvature of the adjoint sequence is a $d_{A}$ -cocycle whose class is connection-independent — so cohomology and integrability are dual to one another. The van Est map generalises the classical Lie-group-to-Lie-algebra comparison and is the bridge by which this complex feeds the Crainic-Fernandes theory, while the modular class in $H^{1} (A)$ marks where the secondary invariants begin; this pattern recurs whenever an infinitesimal symmetry algebra is dualised into a differential.

Full proof set Master

Proposition (Recovery of de Rham and Chevalley-Eilenberg cohomology). For $A = T M$ the algebroid de Rham complex is the ordinary de Rham complex and $H^{∙} (T M) = H_{dR}^{∙} (M)$ ; for $A = g$ a Lie algebra over a point the complex is the Chevalley-Eilenberg complex and $H^{∙} (g) = H_{CE}^{∙} (g)$ .

Proof. For $A = T M$ , $ρ = id$ and the bracket is the Lie bracket of vector fields. Substituting into the Koszul formula reproduces the invariant Cartan formula for $d$ on $Ω^{∙} (M)$ , so $d_{A} = d$ and the cohomologies agree. For $A = g$ over a point, $ρ = 0$ kills the first sum, leaving $(d_{A} ω) (X_{0}, \dots, X_{p}) = \sum_{i < j} (- 1)^{i + j} ω ([X_{i}, X_{j}], \dots)$ , which is the Chevalley-Eilenberg differential on $Λ^{∙} g^{*}$ ^{[Chevalley-Eilenberg 1948]}. Its cohomology is $H_{CE}^{∙} (g)$ by definition. The two computations are the anchor-only and bracket-only specialisations of the same formula. $□$

Proposition (Connection-independence of the obstruction class). Let $A$ be a transitive Lie algebroid with adjoint sequence $0 \to ker ρ \to A \to T M \to 0$ and a connection (splitting) $σ : T M \to A$ with curvature $Ω_{σ} (X, Y) = [σ X, σ Y] - σ [X, Y] \in Γ (ker ρ)$ . Then $Ω_{σ}$ is a $d_{A}$ -cocycle in $Ω^{2} (A; ker ρ)$ and its class $[Ω_{σ}] \in H^{2} (A; ker ρ)$ is independent of $σ$ .

Proof. Restricting $d_{A}$ to $T M$ -arguments via the splitting, $Ω_{σ}$ is the curvature of the $A$ -connection induced on $ker ρ$ , and the Bianchi identity for that connection reads $d_{A}^{\nabla} Ω_{σ} = 0$ , so $Ω_{σ}$ is closed in the coefficient complex. Two splittings differ by a bundle map $τ : T M \to ker ρ$ , i.e. $σ^{'} = σ + τ$ . A direct expansion gives $Ω_{σ^{'}} - Ω_{σ} = d_{A}^{\nabla} τ$ , the coefficient differential of the $1$ -cochain $τ \in Ω^{1} (A; ker ρ)$ . Hence $[Ω_{σ^{'}}] = [Ω_{σ}]$ in $H^{2} (A; ker ρ)$ , and the class depends only on $A$ . This is the cohomological home of the integrability obstruction of 03.04.18. $□$

Proposition (Poisson cohomology is the cotangent-algebroid cohomology). *For a Poisson manifold $(M, π)$ with cotangent algebroid $A = T^{*} M$ , anchor $π^{♯}$ , and Koszul bracket, the algebroid differential is $d_{A} = - [π, \cdot]_{Schouten}$ on multivector fields, and $H^{∙} (A) = H_{π}^{∙} (M)$ is Poisson cohomology.*

Proof. Under the anchor $π^{♯}$ , a $1$ -form $α$ maps to the Hamiltonian field $π^{♯} α$ , and the Koszul bracket $[α, β]_{π} = L_{π^{♯} α} β - L_{π^{♯} β} α - d (π (α, β))$ makes $T^{*} M$ a Lie algebroid. Dualising, $Ω^{p} (A) = Γ (Λ^{p} T M)$ are multivector fields. Substituting $ρ = π^{♯}$ and the Koszul bracket into the Koszul formula, and using the defining property of the Schouten-Nijenhuis bracket, the resulting operator on $Λ^{∙} T M$ is $X \mapsto - [π, X]$ , which squares to zero exactly because $[π, π] = 0$ is the Jacobi identity for $π$ . Its cohomology is by definition Lichnerowicz-Poisson cohomology $H_{π}^{∙} (M)$ ^{[Lichnerowicz 1977]}. So the abstract $d_{A}^{2} = 0 \Leftrightarrow$ Jacobi specialises here to $[π, π] = 0$ . $□$

Connections Master

The Lie algebroid 03.04.16 is the input this unit dualises: its anchor, bracket, and Leibniz rule are exactly the data the Koszul formula consumes, and the equivalence $d_{A}^{2} = 0 \Leftrightarrow$ Jacobi shows that the algebroid axioms and the differential complex are two encodings of one structure. Where that unit stops at the bracket, this one builds the dual differential and its cohomology.

De Rham cohomology 03.04.06 is the special case $A = T M$ with $ρ = id$ : the Koszul formula becomes the invariant Cartan formula for the exterior derivative, so ordinary de Rham cohomology is literally the algebroid cohomology of the tangent algebroid, and every feature of $H^{∙} (A)$ generalises a feature of $H_{dR}^{∙} (M)$ .

Lie algebra (Chevalley-Eilenberg) cohomology 03.04.01 is the special case $A = g$ over a point, where the vanishing anchor strips the formula down to the bracket term and recovers the 1948 differential on $Λ^{∙} g^{*}$ . The algebroid theory is the common generalisation of the de Rham and the Chevalley-Eilenberg complexes, with the base manifold and the structure algebra varying together.

The Pradines integration obstruction 03.04.18 names $H^{2} (A; ker ρ)$ as the home of its class but does not construct the complex; this unit supplies it. The curvature of the adjoint sequence is a $d_{A}$ -cocycle whose connection-independent class is exactly that obstruction, so the cohomology built here is the missing technical object underneath the integration dichotomy.

Representations of Lie algebroids 03.04.23 extend this complex to coefficients: a flat $A$ -connection on a bundle $E$ turns $d_{A}$ into a coefficient differential $d_{A}^{\nabla}$ with cohomology $H^{∙} (A; E)$ , the natural target into which $H^{2} (A; ker ρ)$ and the modular class in $H^{1} (A)$ fit. That forward unit is where the coefficient bundles of this one acquire their flat connections.

Historical & philosophical context Master

Claude Chevalley and Samuel Eilenberg, in their 1948 paper on the cohomology of Lie groups and Lie algebras, introduced the differential on the exterior algebra of the dual of a Lie algebra that now bears their names, recasting the de Rham cohomology of a compact Lie group as a purely algebraic computation on its Lie algebra ^{[Chevalley-Eilenberg 1948]}. Their construction was the prototype: a bracket on a finite-dimensional space induces a square-zero differential on the exterior algebra of the dual, with closedness and exactness governed by the algebra's structure. The same year, de Rham's theorem was being absorbed into the language of sheaves and complexes, and the parallel between "differentiate forms on a manifold" and "differentiate cochains on a Lie algebra" was visible but not yet unified.

The unification came through the Lie algebroid. Jean Pradines' 1960s calculus of differentiable groupoids and Kirill Mackenzie's systematic development in 1987 made the anchor-and-bracket structure first-class, and Mackenzie's Chapter III constructs precisely the de Rham complex of an algebroid, observing that $A = T M$ returns de Rham and $A = g$ returns Chevalley-Eilenberg ^{[Mackenzie 1987]}. André Lichnerowicz's 1977 study of Poisson manifolds had already, in different language, written down the same differential for the cotangent algebroid as the Schouten bracket with the Poisson bivector, giving Poisson cohomology ^{[Lichnerowicz 1977]}. The conceptual payoff arrived with Marius Crainic and Rui Loja Fernandes in 2003, who placed the integration obstruction of a Lie algebroid in $H^{2}$ with isotropy coefficients, completing the arc in which a cohomology theory built from a bracket measures whether the infinitesimal object can be integrated to a global one ^{[Crainic-Fernandes 2003]}.

Bibliography Master

@article{chevalley-eilenberg1948,
  author  = {Chevalley, Claude and Eilenberg, Samuel},
  title   = {Cohomology Theory of Lie Groups and Lie Algebras},
  journal = {Transactions of the American Mathematical Society},
  volume  = {63},
  number  = {1},
  pages   = {85--124},
  year    = {1948}
}

@book{mackenzie1987,
  author    = {Mackenzie, Kirill},
  title     = {Lie Groupoids and Lie Algebroids in Differential Geometry},
  series    = {London Mathematical Society Lecture Note Series},
  volume    = {124},
  publisher = {Cambridge University Press},
  year      = {1987}
}

@book{mackenzie2005,
  author    = {Mackenzie, Kirill C. H.},
  title     = {General Theory of Lie Groupoids and Lie Algebroids},
  series    = {London Mathematical Society Lecture Note Series},
  volume    = {213},
  publisher = {Cambridge University Press},
  year      = {2005}
}

@article{lichnerowicz1977,
  author  = {Lichnerowicz, Andr\'e},
  title   = {Les vari\'et\'es de Poisson et leurs alg\`ebres de Lie associ\'ees},
  journal = {Journal of Differential Geometry},
  volume  = {12},
  number  = {2},
  pages   = {253--300},
  year    = {1977}
}

@article{crainic-fernandes2003,
  author  = {Crainic, Marius and Fernandes, Rui Loja},
  title   = {Integrability of Lie brackets},
  journal = {Annals of Mathematics},
  volume  = {157},
  number  = {2},
  pages   = {575--620},
  year    = {2003}
}

Lie algebroid cohomology — one Chevalley-Eilenberg / Koszul differential $d_{A}$ on $\Gamma(\Lambda^\bullet A^) $w h oses q u a r e - z er oco n d i t i o ni se x a c tl y t h e J a co bii d e n t i t y, u ni f y in g d e R ham co h o m o l o g y ($ A = TM $), L i e - a l g e b r a co h o m o l o g y ($ A = \mathfrak g $), P o i sso n - L i c hn er o w i cz co h o m o l o g y (co t an g e n t a l g e b r o i d), an df o l ia t e d co h o m o l o g y, w i t h t h e in t e g r abi l i t y o b s t r u c t i o n o f [03.04.18] l i v in g in$ H^2(A;\ker\rho)$.*

Prerequisites

03.04.16
03.04.06
03.04.01
03.04.18

Tier anchors

beginner: one differential complex that is simultaneously de Rham cohomology, Lie-algebra cohomology, and Poisson cohomology — depending on which 'algebroid' you feed it
intermediate: Mackenzie — General Theory of Lie Groupoids and Lie Algebroids Ch. III §3 (the Chevalley-Eilenberg / de Rham complex of a Lie algebroid)
master: Mackenzie — General Theory of Lie Groupoids and Lie Algebroids Ch. III §3; Chevalley-Eilenberg — Cohomology theory of Lie groups and Lie algebras (Trans. AMS 63, 1948); Crainic-Fernandes — Integrability of Lie brackets (Ann. Math. 157, 2003)

References

Mackenzie — General Theory of Lie Groupoids and Lie Algebroids · Ch. III §3, the Lie algebroid de Rham complex and its cohomology
Chevalley-Eilenberg — Cohomology theory of Lie groups and Lie algebras · Transactions of the AMS 63 (1948), the differential on the exterior algebra of the dual
Crainic-Fernandes — Integrability of Lie brackets · Annals of Mathematics 157 (2003), the integration obstruction as a cohomology class
Lichnerowicz — Les variétés de Poisson et leurs algèbres de Lie associées · J. Diff. Geom. 12 (1977), the Poisson cohomology differential as the cotangent-algebroid case
Mackenzie — Lie Groupoids and Lie Algebroids in Differential Geometry · LMS Lecture Notes 124 (1987), Ch. III, algebroid cohomology and the adjoint sequence

Estimated time

beginner: 18m
intermediate: 50m
master: 85m