03.04.23 · modern-geometry / differential-forms

Representations of a Lie algebroid and flat A-connections

shipped3 tiersLean: nonepending prereqs

Anchor (Master): Mackenzie — General Theory of Lie Groupoids and Lie Algebroids Ch. III §2 + Ch. V; Crainic — Differentiable and algebroid cohomology, van Est isomorphisms, and characteristic classes (Comment. Math. Helv. 78, 2003); Abad-Crainic — Representations up to homotopy of Lie algebroids (J. reine angew. Math. 663, 2012)

Intuition Beginner

A Lie algebroid is a system of allowed directions over a base space, with two pieces of data: a shadow rule that sends each direction down to an honest motion of the base, and a bracket that says how two directions interact. A representation asks this system to act on an extra bundle of vectors riding over the base, telling each direction how to nudge those vectors as you move.

The nudging rule is called an A-connection. It takes a direction in the algebroid and a section of the bundle, and returns a new section: the rate at which the bundle's vectors change as you flow along that direction. There are two natural demands. First, the rule should be sensible when you scale, which forces a correction term tied to the shadow rule.

Second, and this is the heart of it, the action should respect the bracket. Moving along direction one and then two, versus two and then one, should differ exactly by moving along their bracket. When that holds the connection is called flat, and a flat A-connection is what we mean by a representation. Flatness is the statement that the action has no hidden curvature, no surprise twist when you go around a small loop.

This one idea unifies several older ones. When the algebroid is just the tangent directions of a space, a flat action is a flat bundle, the kind that carries a representation of loops in the space. When the algebroid is a single Lie algebra sitting over a point, a flat action is an ordinary Lie-algebra representation.

Visual Beginner

Picture the base as a flat sheet, and above each point a small fibre, a stack of arrows that can be rotated. A direction in the algebroid is drawn as an arrow on the sheet together with an instruction dial telling the stack how to spin as you slide that way.

Now trace a tiny square: slide along direction one, then two, then back along one, then two. The fibre's arrows get spun four times. If they come back exactly to where they started, the action is flat and the curvature gauge reads zero. If they come back rotated, there is curvature, and the leftover rotation is what the curvature measures. A representation is the flat case, where every such loop closes.

The side panel shows the bookkeeping behind flatness. Doing direction one then two, compared with two then one, never quite cancels on its own; the mismatch is supposed to be accounted for by sliding along the bracket of the two directions. Flat means the mismatch and the bracket-slide agree perfectly, with nothing left over.

Worked example Beginner

Take the base to be a circle, and over it the simplest non-flat-looking setup: a bundle whose fibre is the plane, two real numbers stacked over each point. Let the algebroid be the tangent directions of the circle, so the only direction is "go around," and its shadow is itself.

A connection here is a rule for spinning the plane as you walk around the circle. Pick the rule that rotates the plane at a steady rate, completing a quarter turn over the full loop. Walk once around: a vector that started pointing east now points north. It did not return to itself, so going around the loop produced a real rotation.

Because the circle has only one direction, there is no second direction to bracket against, so flatness here just asks whether walking the loop returns every vector unchanged. The steady quarter-turn rule fails that, so it is a connection but not a flat one, not a representation. To get a representation, choose the rule that does no spinning at all: walk around and every vector comes back exactly. That zero rule is flat, hence a genuine representation of the tangent algebroid of the circle.

What this shows: a connection is any consistent spinning rule, and only the flat ones, the ones with no leftover rotation around loops, count as representations. The flat rules on the circle's tangent algebroid match the ways loops in the circle can act on a plane, which is the flat-bundle picture in miniature.

Check your understanding Beginner

Formal definition Intermediate+

Let $A \to M$ be a Lie algebroid 03.04.16 with anchor $ρ : A \to T M$ and Lie bracket $[\cdot, \cdot]$ on $Γ (A)$ , and let $E \to M$ be a vector bundle. An $A$ -connection on $E$ is an $R$ -bilinear map $$ \nabla : \Gamma(A) \times \Gamma(E) \to \Gamma(E), \qquad (X,s) \mapsto \nabla_X s, $$ that is $C^{\infty} (M)$ -linear in the algebroid slot and satisfies the anchored Leibniz rule in the bundle slot: $$ \nabla_{fX} s = f,\nabla_X s, \qquad \nabla_X(f s) = f,\nabla_X s + \big(\rho(X)f\big),s, $$ for all $f \in C^{\infty} (M)$ , $X \in Γ (A)$ , $s \in Γ (E)$ . The Leibniz term is governed by the anchor, so the derivative of a function picked up by $\nabla$ is $ρ (X) f$ , not the bare directional derivative of an ordinary connection 03.05.04. The curvature of $\nabla$ is $$ R^\nabla(X,Y),s ;=; \nabla_X \nabla_Y s - \nabla_Y \nabla_X s - \nabla_{[X,Y]} s, $$ which one checks is $C^{\infty} (M)$ -bilinear in $X, Y$ and $C^{\infty} (M)$ -linear in $s$ , hence a section $R^{\nabla} \in Γ (Λ^{2} A^{*} \otimes End E)$ .

A representation of the Lie algebroid $A$ on $E$ , equivalently a flat $A$ -connection, is an $A$ -connection with $R^{\nabla} = 0$ . Spelled out, flatness is the operator identity $$ \nabla_{[X,Y]} = [\nabla_X, \nabla_Y] = \nabla_X \nabla_Y - \nabla_Y \nabla_X, $$ so $X \mapsto \nabla_{X}$ is a Lie-algebra homomorphism from $Γ (A)$ to the first-order operators on $Γ (E)$ . The sign convention is the standard one of Mackenzie and Crainic: curvature is the failure of $\nabla$ to be a bracket homomorphism, and a flat connection is precisely a bracket homomorphism, matching the flat-splitting picture of 03.05.23 ^{[Mackenzie Ch. III §2]}.

The reformulation that makes the structure transparent uses the derivation algebroid (or Atiyah algebroid of $E$ ), written $D (E)$ or $gl (E)$ . Its sections are the covariant differential operators: $R$ -linear maps $Δ : Γ (E) \to Γ (E)$ for which there exists a vector field $σ (Δ) \in Γ (T M)$ , the symbol, with $Δ (f s) = f Δ s + (σ (Δ) f) s$ . These are a vector bundle $D (E)$ ; the symbol is a bundle map $σ : D (E) \to T M$ , and the commutator of operators is a Lie bracket on $Γ (D (E))$ . With anchor $σ$ and this bracket, $D (E)$ is a transitive Lie algebroid sitting in the exact sequence $$ 0 \longrightarrow \operatorname{End}(E) \longrightarrow D(E) \xrightarrow{\ \sigma\ } TM \longrightarrow 0, $$ whose kernel is the zeroth-order operators $End (E)$ . An $A$ -connection on $E$ is exactly a bundle map $\nabla : A \to D (E)$ over $M$ with $σ \circ \nabla = ρ$ , that is, a lift of the anchor; the connection is flat — a representation — if and only if $\nabla$ is a morphism of Lie algebroids $A \to D (E)$ . This is the definition that generalises cleanly and that the rest of the unit uses ^{[Mackenzie Ch. III §2]}.

Counterexamples to common slips

A representation is not a single bracket-homomorphism into vector fields: the anchored Leibniz rule means $\nabla_{X}$ acts on $E$ , with symbol $ρ (X)$ , so the target is $D (E)$ , not $Γ (T M)$ .
Flatness $R^{\nabla} = 0$ is genuinely stronger than tensoriality of the curvature: every $A$ -connection has a tensorial curvature; only the representations have it vanish.
The adjoint action of $A$ on itself is not a representation in this sense — the naive formula has the wrong tensoriality — and is replaced by a representation up to homotopy, treated below.

Key theorem with proof Intermediate+

Theorem (Recovery of the boundary cases). Let $A \to M$ be a Lie algebroid and $E \to M$ a vector bundle, with $\nabla$ a flat $A$ -connection on $E$ . (i) If $A = T M$ with $ρ = id$ , then $\nabla$ is an ordinary flat linear connection on $E$ , and over a connected $M$ its parallel transport is a representation of the fundamental group $π_{1} (M)$ on a fibre $E_{x}$ ; conversely every such representation arises this way. (ii) If $M = {pt}$ and $A = g$ is a Lie algebra, then $\nabla$ is exactly a representation of $g$ on the vector space $E$ .

Proof. For (i) take $A = T M$ , $ρ = id$ . The anchored Leibniz rule becomes $\nabla_{X} (f s) = f \nabla_{X} s + (X f) s$ , the rule for a linear connection 03.05.04, and $C^{\infty}$ -linearity in $X$ is automatic, so $\nabla$ is an ordinary connection. Its curvature $R^{\nabla} (X, Y) = \nabla_{X} \nabla_{Y} - \nabla_{Y} \nabla_{X} - \nabla_{[X, Y]}$ is the usual curvature tensor, and flatness $R^{\nabla} = 0$ is vanishing of that tensor. On a connected manifold a flat connection has path-independent parallel transport up to homotopy: transport around a contractible loop is the identity because the curvature integrand over a spanning disc vanishes, so transport descends to a homomorphism from homotopy classes of loops based at $x$ , namely $π_{1} (M, x) \to G L (E_{x})$ , the holonomy representation. Conversely, given a representation $μ : π_{1} (M, x) \to G L (V)$ , form the associated bundle $E = M \times_{π_{1}} V$ over the universal cover $M$ ; its tautological flat connection has holonomy $μ$ . This is the Riemann-Hilbert correspondence between flat bundles and $π_{1}$ -representations ^{[Mackenzie Ch. III §2]}.

For (ii) take $M$ a point, so $T M = 0$ and $ρ = 0$ . The anchored Leibniz term $ρ (X) f$ vanishes, $C^{\infty} (M) = R$ , and $E$ is a single vector space. An $A$ -connection is then an $R$ -bilinear map $g \times E \to E$ , $(X, v) \mapsto \nabla_{X} v$ , with no Leibniz constraint, that is, a linear action of $g$ on $E$ . Its curvature is $R^{\nabla} (X, Y) = \nabla_{X} \nabla_{Y} - \nabla_{Y} \nabla_{X} - \nabla_{[X, Y]}$ , and flatness reads $\nabla_{[X, Y]} = \nabla_{X} \nabla_{Y} - \nabla_{Y} \nabla_{X}$ , the defining identity of a Lie-algebra representation $g \to gl (E)$ . So flat $g$ -connections are exactly $g$ -representations. $□$

Bridge. This recovery builds toward the unified coefficient theory of the next section: the foundational reason flat bundles and Lie-algebra representations look so different at first sight is that they are the two extreme anchors, $ρ = id$ and $ρ = 0$ , of one definition, and the morphism-into- $D (E)$ picture is exactly the statement that interpolates between them. The holonomy of part (i) is dual to the integration story: a representation of $A$ is the infinitesimal shadow of a representation of an integrating groupoid, so this is exactly the flat-connection special case of 03.05.23 read at the level of an arbitrary algebroid rather than an Atiyah algebroid. Putting these together, the central insight is that "representation" and "flat connection" are one notion, and it generalises the classical $π_{1}$ -action and the classical $g$ -action simultaneously; this same identification appears again in the cohomology with coefficients $H^{∙} (A; E)$ , where the flat connection is precisely the twist needed to differentiate $E$ -valued algebroid forms.

Exercises Intermediate+

Exercise 4 (medium, symbolic).

Let $A = g ⋉ M$ be the action algebroid of a $g$ -action on $M$ , with anchor $ρ (X) = X_{M}$ the fundamental vector field. Describe what a representation of $A$ on the product bundle $E = M \times V$ is, concretely.

Hint

Sections of $A$ are $g$ -valued functions; evaluate $\nabla$ on constant sections.

Answer

A flat $A$ -connection assigns to each $X \in g$ (as a constant section) an operator $\nabla_{X} = X_{M} + μ (X)$ on $Γ (M \times V) = C^{\infty} (M, V)$ , where $μ (X) \in C^{\infty} (M, End V)$ . The anchored Leibniz rule is automatic from $X_{M}$ , and flatness $\nabla_{[X, Y]} = [\nabla_{X}, \nabla_{Y}]$ becomes $μ ([X, Y]) = X_{M} μ (Y) - Y_{M} μ (X) + [μ (X), μ (Y)]$ . This is precisely the cocycle condition for an equivariant family of representations: a representation of the action algebroid is a $g$ -equivariant structure on $M \times V$ , generalising a flat $g$ -bundle with action data.

Exercise 5 (medium, proof).

Prove that an $A$ -connection $\nabla$ on $E$ is flat if and only if the corresponding bundle map $\nabla : A \to D (E)$ (covering $ρ$ via the symbol) is a morphism of Lie algebroids.

Hint

A bundle map over $M$ that intertwines anchors is an algebroid morphism iff it preserves brackets of sections; compare $\nabla_{[X, Y]}$ with $[\nabla_{X}, \nabla_{Y}]_{D (E)}$ .

Answer

By construction $σ \circ \nabla = ρ$ , so $\nabla$ intertwines the anchors. For a base-preserving bundle map this makes it an algebroid morphism exactly when it preserves brackets of sections. The bracket in $Γ (D (E))$ is the operator commutator, so the morphism condition reads $\nabla_{[X, Y]} = [\nabla_{X}, \nabla_{Y}]$ in $D (E)$ , i.e. $\nabla_{[X, Y]} s = \nabla_{X} \nabla_{Y} s - \nabla_{Y} \nabla_{X} s$ for all $s$ . That is $R^{\nabla} (X, Y) = 0$ . (One checks $C^{\infty}$ -linearity is preserved automatically since both sides agree on $f X$ by $C^{\infty}$ -linearity of $\nabla$ in the $A$ -slot.) Hence flat $⟺$ algebroid morphism. $□$

Exercise 6 (hard, proof).

Given a representation $(E, \nabla)$ of $A$ , define $d_{\nabla} : Γ (Λ^{k} A^{*} \otimes E) \to Γ (Λ^{k + 1} A^{*} \otimes E)$ by the Chevalley-Eilenberg formula with $\nabla$ in place of the anchor action. Prove $d_{\nabla}^{2} = 0$ , so that $H^{∙} (A; E)$ is defined 03.04.22.

Hint

$d_{\nabla}^{2}$ is a tensorial operator whose value is built from $R^{\nabla}$ ; flatness makes it vanish.

Answer

The twisted differential is $$ (d_\nabla \omega)(X_0,\dots,X_k) = \sum_i (-1)^i \nabla_{X_i}\omega(\dots\widehat{X_i}\dots) + \sum_{i<j}(-1)^{i+j}\omega([X_i,X_j],\dots\widehat{X_i}\dots\widehat{X_j}\dots). $$ A direct expansion of $d_{\nabla}^{2}$ collects the terms with two $\nabla$ 's and one bracket; using the Jacobi identity for $[\cdot, \cdot]$ on the bracket-only terms and antisymmetry, every contribution cancels except those proportional to $\nabla_{X_{i}} \nabla_{X_{j}} - \nabla_{X_{j}} \nabla_{X_{i}} - \nabla_{[X_{i}, X_{j}]}$ , which is $R^{\nabla} (X_{i}, X_{j})$ . Thus $d_{\nabla}^{2} ω$ is a tensorial expression linear in $R^{\nabla}$ . With $\nabla$ flat, $R^{\nabla} = 0$ , so $d_{\nabla}^{2} = 0$ . The cohomology of $(Γ (Λ^{∙} A^{*} \otimes E), d_{\nabla})$ is $H^{∙} (A; E)$ , the algebroid cohomology with coefficients in the representation $E$ , extending the constant-coefficient complex of 03.04.22. $□$

Exercise 7 (hard, proof).

Let $A$ be transitive with isotropy bundle $L = ker ρ$ . Show that the fibrewise adjoint action $ad : Γ (A) \times Γ (L) \to Γ (L)$ , $ad_{X} (ξ) = [X, ξ]$ , is a representation of $A$ on $L$ , and identify $H^{0} (A; L)$ .

Hint

First check $[X, ξ] \in Γ (L)$ for $ξ \in Γ (L)$ using that $ρ$ is a bracket homomorphism; then verify Leibniz and flatness from the Jacobi identity.

Answer

For $ξ \in Γ (L)$ , $ρ [X, ξ] = [ρX, ρ ξ] = [ρX, 0] = 0$ , so $[X, ξ] \in Γ (L)$ and $ad_{X}$ preserves $L$ . The anchored Leibniz rule $ad_{X} (f ξ) = [X, f ξ] = f [X, ξ] + (ρ (X) f) ξ$ holds by the algebroid Leibniz rule, and $C^{\infty}$ -linearity $ad_{f X} ξ = [f X, ξ] = f [X, ξ] - (ρ (ξ) f) X = f [X, ξ]$ since $ρ (ξ) = 0$ . So $ad$ is an $A$ -connection on $L$ . Flatness is the Jacobi identity: $ad_{[X, Y]} ξ = [[X, Y], ξ] = [X, [Y, ξ]] - [Y, [X, ξ]] = (ad_{X} ad_{Y} - ad_{Y} ad_{X}) ξ$ . Hence $L$ is a genuine representation (this is the adjoint representation on the isotropy, defined precisely because $L$ is a subalgebroid with $ρ ∣_{L} = 0$ ). Finally $H^{0} (A; L) = {ξ \in Γ (L) : [X, ξ] = 0 \forall X}$ , the invariant isotropy sections, the centre of the isotropy as an $A$ -module.

Advanced results Master

The coefficient complex and $H^{∙} (A; E)$ . A representation $(E, \nabla)$ twists the Lie-algebroid de Rham complex 03.04.22 into the complex $(Ω^{∙} (A; E), d_{\nabla})$ with $Ω^{k} (A; E) = Γ (Λ^{k} A^{*} \otimes E)$ and the Chevalley-Eilenberg differential built from $\nabla$ and the bracket. Flatness is exactly $d_{\nabla}^{2} = 0$ , and the cohomology $H^{∙} (A; E)$ is the Lie-algebroid cohomology with coefficients in $E$ . The two boundary cases reproduce classical theories: $A = T M$ gives the de Rham cohomology of $M$ with coefficients in a flat bundle (twisted de Rham, the home of the de Rham side of Riemann-Hilbert), and $A = g$ over a point gives the Chevalley-Eilenberg cohomology $H^{∙} (g; E)$ of a Lie-algebra module. Low degrees carry the familiar meaning: $H^{0} (A; E)$ is the invariant sections, $H^{1} (A; E)$ classifies derivations modulo inner ones and governs first-order deformations ^{[Crainic 2003]}.

Characteristic classes of a representation. A representation $E$ of $A$ , together with a choice of (possibly non-flat) auxiliary connection, produces secondary characteristic classes in $H^{∙} (A; R)$ by a Chern-Weil-type construction adapted to algebroids; for $A = T M$ these recover the classical characteristic classes of flat bundles, and the simplest one, attached to the line bundle $Λ^{top} E$ , is the modular class of the representation. For the canonical representation of the cotangent algebroid $T^{*} M$ of a Poisson manifold on $Λ^{top} T^{*} M$ , this is the modular class of the Poisson manifold, the obstruction to a $\nabla$ -invariant volume ^{[Fernandes 2002]}.

Poisson modules. For the cotangent algebroid $T^{*} M$ of a Poisson manifold $(M, π)$ , a representation is a flat $T^{*} M$ -connection, called a Poisson module: a vector bundle $E$ with a contravariant flat connection $\nabla_{α} s$ along one-forms $α$ , with anchor the sharp map $π^{♯}$ . The plain line bundle with the canonical contravariant connection has cohomology the Poisson cohomology with coefficients, and the modular class above lives there. Poisson modules are the natural target for the cotangent-algebroid cohomology and integrate (when $T^{*} M$ integrates) to representations of the symplectic groupoid ^{[Mackenzie Ch. V]}.

The adjoint and representations up to homotopy. There is no canonical representation of $A$ on $A$ itself that deserves the name "adjoint": the naive formula $\nabla_{X} Y = [X, Y]$ fails $C^{\infty} (M)$ -linearity in $X$ , since $[f X, Y] = f [X, Y] - (ρ (Y) f) X$ . The correct object is a representation up to homotopy on the $2$ -term complex $A ρ T M$ : a degree-one superconnection $D$ on $A \oplus T M [1]$ whose square is zero up to coherent homotopy, encoded by the anchor in degree zero, a chosen connection $\nabla$ on $A$ in degree one, and the curvature of $\nabla$ as the degree-two homotopy. Its cohomology, deformation cohomology $H_{def}^{∙} (A)$ , controls deformations of the algebroid bracket and replaces the ill-defined $H^{∙} (A; A)$ . The construction is independent of the auxiliary $\nabla$ up to isomorphism of representations up to homotopy, and for $A = T M$ it recovers the (evidently flat) adjoint of the tangent algebroid ^{[Abad-Crainic 2012]}.

Synthesis. A representation of a Lie algebroid is a flat $A$ -connection on a vector bundle, equivalently a Lie-algebroid morphism $A \to D (E)$ into the derivation algebroid; this single definition is the foundational reason that flat vector bundles, Lie-algebra representations, and Poisson modules are one theory, recovered as the anchors $ρ = id$ , $ρ = 0$ , and $ρ = π^{♯}$ . Putting these together, the flat connection is exactly the twist that makes the $E$ -valued algebroid de Rham complex square to zero, so representations are the natural coefficient systems for the cohomology $H^{∙} (A; E)$ of 03.04.22, and the modular and secondary characteristic classes are dual to the choice of an invariant section or volume. This generalises the classical correspondence between $π_{1}$ -representations and flat bundles, the central insight being that holonomy of a flat $A$ -connection is the infinitesimal shadow of a groupoid representation, exactly the flat-splitting picture of 03.05.23 read for an arbitrary algebroid. The adjoint is the one case the definition cannot hold: it is a representation up to homotopy on $A \to T M$ , and that failure is itself structural, since deformation cohomology rather than $H^{∙} (A; A)$ is what governs how the bracket can move.

Full proof set Master

Proposition (An $A$ -connection always exists; flatness is the constraint). For any Lie algebroid $A \to M$ and vector bundle $E \to M$ there exists an $A$ -connection on $E$ . The flat ones, when nonempty, form a torsor under closed $E$ -twisted algebroid $1$ -forms modulo a gauge condition.

Proof. Choose an ordinary linear connection $\nabla^{0}$ on $E$ 03.05.04, with covariant derivative along vector fields. Define $\nabla_{X} := \nabla_{ρ (X)}^{0}$ . Then $\nabla_{f X} = \nabla_{ρ (f X)}^{0} = \nabla_{f ρ (X)}^{0} = f \nabla_{ρ (X)}^{0} = f \nabla_{X}$ , and $\nabla_{X} (f s) = \nabla_{ρ (X)}^{0} (f s) = f \nabla_{ρ (X)}^{0} s + (ρ (X) f) s = f \nabla_{X} s + (ρ (X) f) s$ , so $\nabla$ is an $A$ -connection. Existence holds with no obstruction. For the torsor statement, the difference of two $A$ -connections is $C^{\infty} (M)$ -linear in both slots, hence a section $θ \in Ω^{1} (A; End E) = Γ (A^{*} \otimes End E)$ . The curvature of $\nabla + θ$ is $R^{\nabla + θ} = R^{\nabla} + d_{\nabla} θ + \frac{1}{2} [θ, θ]$ , the algebroid Maurer-Cartan expression. Fixing one flat $\nabla$ , the flat connections are those $θ$ with $d_{\nabla} θ + \frac{1}{2} [θ, θ] = 0$ ; for the abelian (line-bundle) case this is the affine condition $d_{\nabla} θ = 0$ , exhibiting the set of flat connections as a torsor under closed twisted $1$ -forms. $□$

Proposition (Morphism reformulation is an equivalence of categories). The assignment $(E, \nabla) \mapsto (\nabla : A \to D (E))$ is an isomorphism between flat $A$ -connections on $E$ and Lie-algebroid morphisms $A \to D (E)$ over $id_{M}$ covering $ρ$ along the symbol. It is functorial: a bundle map $E \to E^{'}$ intertwining the connections corresponds to a morphism compatible with $D (E) \to D (E^{'})$ .

Proof. Given $\nabla$ , set $\nabla (X) := \nabla_{X} \in Γ (D (E))$ , a covariant differential operator with symbol $ρ (X)$ ; $C^{\infty} (M)$ -linearity of $\nabla$ in $X$ makes $X \mapsto \nabla_{X}$ a bundle map $A \to D (E)$ , and $σ (\nabla_{X}) = ρ (X)$ says it covers $ρ$ . By Exercise 5 it is an algebroid morphism iff $R^{\nabla} = 0$ . Conversely a bundle map $Φ : A \to D (E)$ with $σ Φ = ρ$ defines $\nabla_{X} s := Φ (X) s$ ; the symbol condition gives the anchored Leibniz rule and $C^{\infty}$ -linearity in $X$ is the bundle-map property, so $\nabla$ is an $A$ -connection, flat iff $Φ$ preserves brackets. The two assignments are mutually inverse. Functoriality: a vector-bundle map $ψ : E \to E^{'}$ induces $D (ψ)$ on operators that conjugate through $ψ$ ; $ψ$ intertwines $\nabla, \nabla^{'}$ exactly when $D (ψ) \circ \nabla = \nabla^{'}$ , the compatibility of morphisms. $□$

Proposition (Pullback and tensor of representations). Representations of $A$ form a tensor category with duals: if $(E, \nabla^{E})$ and $(F, \nabla^{F})$ are representations, so are $E \oplus F$ , $E \otimes F$ with $\nabla_{X}^{E \otimes F} = \nabla_{X}^{E} \otimes 1 + 1 \otimes \nabla_{X}^{F}$ , $E^ $w i t h$ \langle \nabla^{E^}_X\xi, s\rangle = \rho(X)\langle\xi,s\rangle - \langle\xi,\nabla^E_X s\rangle $, an d$ \operatorname{Hom}(E,F) $; t h e u ni t i s t h ec an o ni c a l r e p r ese n t a t i o n o n$ M\times\mathbb R $w i t h$ \nabla_X = \rho(X)$.

Proof. Each formula is checked to be an $A$ -connection and then flat. For the tensor product, the anchored Leibniz rule on $E \otimes F$ follows from the product rule and the two anchored Leibniz rules, and the curvature is $R^{E \otimes F} (X, Y) = R^{E} (X, Y) \otimes 1 + 1 \otimes R^{F} (X, Y)$ by expanding the commutator and using that $\nabla^{E} \otimes 1$ and $1 \otimes \nabla^{F}$ commute; both summands vanish, so $E \otimes F$ is flat. For the dual, differentiating the pairing $⟨ ξ, s ⟩ \in C^{\infty} (M)$ along $ρ (X)$ and imposing the Leibniz rule forces the stated formula; its curvature is $R^{E^{*}} (X, Y) = - R^{E} (X, Y)^{*}$ , which vanishes when $R^{E}$ does. The canonical representation $\nabla_{X} = ρ (X)$ is flat because $ρ$ is a bracket homomorphism, $ρ ([X, Y]) = [ρX, ρ Y]$ , giving $\nabla_{[X, Y]} = [\nabla_{X}, \nabla_{Y}]$ . $Hom (E, F) ≅ E^{*} \otimes F$ inherits flatness, completing the closed tensor structure. $□$

Connections Master

A Lie algebroid 03.04.16 is the structure being represented: its anchor and bracket are exactly the data an $A$ -connection must respect, and the anchored Leibniz rule together with flatness are the algebroid analogue of a module structure. The bracket-homomorphism property of the anchor, established there, is precisely what makes the canonical representation $\nabla_{X} = ρ (X)$ flat, so this unit is the representation theory of the object that one defines.

Algebroid cohomology 03.04.22 is where a representation lands: a flat $A$ -connection is the twist that upgrades the constant-coefficient de Rham complex of the algebroid into the coefficient complex $(Ω^{∙} (A; E), d_{\nabla})$ computing $H^{∙} (A; E)$ . The condition $d_{\nabla}^{2} = 0$ is identical to flatness, so coefficients in the sense of that unit and representations in the sense of this one are the same data, and the adjoint's failure to be a representation is exactly why $H^{∙} (A; A)$ is replaced by deformation cohomology.

The connection-as-splitting picture 03.05.23 is the principal-bundle special case: there the algebroid is the Atiyah algebroid of $P$ , and a flat connection is a flat splitting of the Atiyah sequence, equivalently a bracket-homomorphism into the algebroid; the present unit takes that flat-equals-bracket-homomorphism identity and reads it for an arbitrary algebroid acting on an arbitrary associated bundle, recovering the principal flat connection when $A = At (P)$ and $E$ is associated to $P$ .

A vector bundle connection 03.05.04 is the boundary case $A = T M$ : an ordinary linear connection is a $T M$ -connection, and a representation of $T M$ is a flat linear connection, the flat-bundle/local-system side of the Riemann-Hilbert correspondence. The anchored Leibniz rule degenerates to the ordinary Leibniz rule precisely because the anchor of $T M$ is the identity, which is the sense in which this unit generalises the classical theory of connections.

A Lie algebra 03.04.01 supplies the other boundary case: over a single point a representation of the algebroid $g$ is an ordinary $g$ -module, and the algebroid coefficient cohomology becomes Chevalley-Eilenberg cohomology with coefficients. The two extremes, $ρ = id$ and $ρ = 0$ , sit at opposite ends of one definition, and every algebroid in between interpolates between flat bundles and Lie-algebra modules.

Historical & philosophical context Master

The notion of a flat connection along a Lie algebroid is implicit in Jean Pradines' 1960s formulation of differentiable groupoids and their infinitesimal counterparts, but it was Kirill Mackenzie who, in his 1987 lecture notes and the 2005 monograph, fixed the definition of a representation of a Lie algebroid as a flat $A$ -connection and developed the transitive theory in which the isotropy adjoint and the Atiyah sequence appear ^{[Mackenzie Ch. III §2]}. The recognition that these representations are the coefficient systems for an algebroid de Rham cohomology, with $H^{∙} (A; E)$ generalising both flat-bundle de Rham cohomology and Chevalley-Eilenberg cohomology, was systematised by Marius Crainic in 2003, who also established van Est isomorphisms relating algebroid cohomology to the differentiable cohomology of an integrating groupoid and constructed characteristic classes of representations ^{[Crainic 2003]}.

The obstruction to representing an algebroid on itself was clarified later. Camilo Arias Abad and Marius Crainic, in work published in 2012, introduced representations up to homotopy, showing that the adjoint of a Lie algebroid is naturally a flat representation up to homotopy on the $2$ -term complex $A \to T M$ rather than an honest representation, and that its cohomology is the deformation cohomology governing the algebroid's brackets ^{[Abad-Crainic 2012]}. Alfonso Gracia-Saz and Rajan Mehta gave a parallel double-vector-bundle (VB-algebroid) account of the same superconnection data ^{[Gracia-Saz-Mehta 2010]}. On the Poisson side, Rui Loja Fernandes defined the secondary characteristic classes and the modular class of a Lie algebroid representation, identifying the modular class of a Poisson manifold as that of the canonical representation of its cotangent algebroid ^{[Fernandes 2002]}.

Bibliography Master

@book{mackenzie2005rep,
  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{crainic2003,
  author  = {Crainic, Marius},
  title   = {Differentiable and algebroid cohomology, van Est isomorphisms, and characteristic classes},
  journal = {Commentarii Mathematici Helvetici},
  volume  = {78},
  number  = {4},
  pages   = {681--721},
  year    = {2003}
}

@article{abad-crainic2012,
  author  = {Arias Abad, Camilo and Crainic, Marius},
  title   = {Representations up to homotopy of Lie algebroids},
  journal = {Journal f\"ur die reine und angewandte Mathematik},
  volume  = {663},
  pages   = {91--126},
  year    = {2012}
}

@article{graciasaz-mehta2010,
  author  = {Gracia-Saz, Alfonso and Mehta, Rajan Amit},
  title   = {Lie algebroid structures on double vector bundles and representation theory of Lie algebroids},
  journal = {Advances in Mathematics},
  volume  = {223},
  number  = {4},
  pages   = {1236--1275},
  year    = {2010}
}

@article{fernandes2002,
  author  = {Fernandes, Rui Loja},
  title   = {Lie algebroids, holonomy and characteristic classes},
  journal = {Advances in Mathematics},
  volume  = {170},
  number  = {1},
  pages   = {119--179},
  year    = {2002}
}

@book{mackenzie1987rep,
  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}
}

*Representations of a Lie algebroid and flat A-connections — a representation of $A$ on $E$ is a flat $A$ -connection $\nabla : Γ (A) \times Γ (E) \to Γ (E)$ with anchored Leibniz rule and vanishing curvature $R^{\nabla} (X, Y) = \nabla_{X} \nabla_{Y} - \nabla_{Y} \nabla_{X} - \nabla_{[X, Y]}$ , equivalently a Lie-algebroid morphism $A \to D (E)$ ; it unifies flat bundles / $π_{1}$ -representations ( $A = T M$ ), Lie-algebra representations ( $A = g$ ), and Poisson modules ( $A = T^{*} M$ ), supplies the coefficients for $H^{∙} (A; E)$ , and forces the adjoint to be only a representation up to homotopy on $A \to T M$ (Abad-Crainic).*

Prerequisites

03.04.16 pending
03.05.04 pending
03.05.23 pending
03.04.22 pending

Tier anchors

beginner: the idea of an algebroid acting on a bundle, with flatness meaning the action respects brackets, unifying flat bundles and Lie-algebra representations
intermediate: Mackenzie — General Theory of Lie Groupoids and Lie Algebroids Ch. III §2 (flat connections on a Lie algebroid); Crainic — Differentiable and algebroid cohomology (Comment. Math. Helv. 78)
master: Mackenzie — General Theory of Lie Groupoids and Lie Algebroids Ch. III §2 + Ch. V; Crainic — Differentiable and algebroid cohomology, van Est isomorphisms, and characteristic classes (Comment. Math. Helv. 78, 2003); Abad-Crainic — Representations up to homotopy of Lie algebroids (J. reine angew. Math. 663, 2012)

References

Mackenzie — General Theory of Lie Groupoids and Lie Algebroids · Ch. III §2, representations of a Lie algebroid as flat connections; Ch. V, the adjoint and cohomology with coefficients
Crainic — Differentiable and algebroid cohomology, van Est isomorphisms, and characteristic classes · Comment. Math. Helv. 78 (2003), the A-de Rham complex with coefficients and characteristic classes of representations
Abad-Crainic — Representations up to homotopy of Lie algebroids · J. reine angew. Math. 663 (2012), the adjoint representation as a 2-term representation up to homotopy
Gracia-Saz, Mehta — Lie algebroid structures on double vector bundles and representation theory of Lie algebroids · Adv. Math. 223 (2010), VB-algebroids and the superconnection/2-term picture of the adjoint
Fernandes — Lie algebroids, holonomy and characteristic classes · Adv. Math. 170 (2002), secondary characteristic classes of a representation, the modular class

Estimated time

beginner: 19m
intermediate: 50m
master: 84m