03.04.16 · modern-geometry / differential-forms

Lie algebroid: anchor, bracket, Leibniz law

shipped3 tiersLean: none

Anchor (Master): Mackenzie — General Theory of Lie Groupoids and Lie Algebroids Ch. 3; Crainic-Fernandes — Integrability of Lie brackets (Ann. Math. 157)

Intuition Beginner

A Lie group packages the symmetries of a single object. Its infinitesimal version, the Lie algebra, records the directions you can move away from the identity, together with a way of measuring how two such directions fail to commute. A Lie groupoid is the version where the symmetry runs between pairs of points spread over a space. What is its infinitesimal version?

A Lie algebroid is the answer. At each point of the base space it gives you a vector space of "allowed infinitesimal moves." These moves can be bracketed, the way two vector fields can be bracketed, and each move is allowed to push the base point itself in some direction. The map that reads off how a move pushes the base point is called the anchor.

Why bother? Because many geometric structures are exactly this data in disguise. A flow of vector fields, a group acting on a space, a connection on a bundle, a Poisson structure: each one is a rule for "infinitesimal moves over a base, with a bracket and an anchor." The Lie algebroid is the single object that holds all of them.

Visual Beginner

Picture a curved base surface. Over each point sits a small flat space of arrows — the moves allowed at that point. Some arrows lie flat, pushing nothing; others tilt, dragging their base point along the surface in a definite direction.

The downward shadow of an arrow on the base is its anchor image: the tangent direction it makes the base point move in. When the shadow is zero, the move happens purely "in the fibre" and changes nothing on the base. Two moves can be combined by a bracket, just as two flows on a surface combine into a third flow that measures their failure to commute.

Worked example Beginner

Take the base to be ordinary three-dimensional space, and at each point let the allowed moves be the three rotations: spin about the first axis, the second, the third. Call them $R_{1}$ , $R_{2}$ , $R_{3}$ .

Each rotation pushes a point of space in a definite direction: spinning about the third axis moves a point in a circle around that axis. So the anchor of $R_{3}$ at the point $(x, y, z)$ is the velocity of that circular motion, namely the direction $(- y, x, 0)$ .

How do two of these moves bracket? Spinning about the first axis and then the second is not the same as doing them in the other order; the mismatch is a spin about the third axis. Concretely the bracket of $R_{1}$ and $R_{2}$ is $R_{3}$ , and cycling the labels gives the other two.

What this tells us: the same three rotations appear both as a fixed bracket rule (the bracket of $R_{1}$ and $R_{2}$ is $R_{3}$ ) and as honest motions of points in space (their anchor directions). The anchor and the bracket are two faces of one object, and they fit together consistently. This example is the action algebroid of the rotation group, the simplest case that is neither a plain Lie algebra nor a plain tangent space.

Check your understanding Beginner

Formal definition Intermediate+

Let $M$ be a smooth manifold and $A \to M$ a smooth real vector bundle 03.05.02. Write $Γ (A)$ for the $C^{\infty} (M)$ -module of smooth sections, and $X (M) = Γ (T M)$ for the vector fields on $M$ , whose Lie bracket of vector fields is the operation studied in 03.02.04.

A Lie algebroid structure on $A$ consists of ^{[Mackenzie Ch. 3]}:

an $R$ -bilinear bracket $[\cdot, \cdot] : Γ (A) \times Γ (A) \to Γ (A)$ that is antisymmetric and satisfies the Jacobi identity, making $(Γ (A), [\cdot, \cdot])$ a Lie algebra over $R$ 03.04.01;
a vector-bundle morphism $ρ : A \to T M$ over the identity of $M$ , the anchor, with the induced map on sections still written $ρ : Γ (A) \to X (M)$ ;

subject to the Leibniz rule $$ [X, fY] = f,[X, Y] + \big(\rho(X)f\big),Y $$ for all $X, Y \in Γ (A)$ and $f \in C^{\infty} (M)$ . Here $ρ (X) f$ denotes the directional derivative of $f$ along the vector field $ρ (X)$ , and the symbol $Γ (A)$ denotes the section module; $X (M)$ denotes the vector fields on $M$ , and $⋉$ below denotes the action-algebroid bundle.

The Leibniz rule fixes how the bracket interacts with multiplication by functions: the bracket is a derivation in its second slot, with the function-derivative term governed entirely by the anchor. Antisymmetry transfers the same behaviour to the first slot, $[f X, Y] = f [X, Y] - (ρ (Y) f) X$ . A frequent slip is to demand $C^{\infty} (M)$ -bilinearity of the bracket; the Leibniz term shows the bracket is not $C^{\infty} (M)$ -bilinear unless the anchor vanishes, and it is precisely this defect that lets the anchor be recovered from the bracket.

A second condition often imposed in the definition is that the anchor be a bracket homomorphism, $$ \rho\big([X, Y]\big) = \big[\rho(X), \rho(Y)\big], $$ with the bracket on the right the vector-field bracket on $M$ . This is listed as an axiom in some treatments; the next theorem shows it follows from the Jacobi identity and the Leibniz rule, so it is a consequence rather than an independent requirement.

A non-example sharpens the definition: a vector bundle with an antisymmetric $R$ -bilinear bracket on sections but no compatible anchor — for instance, demanding $[X, f Y] = f [X, Y]$ with no derivative term — forces the bracket to be $C^{\infty} (M)$ -bilinear, hence pointwise, making each fibre a Lie algebra but losing all coupling to the base geometry. That structure is a bundle of Lie algebras, the case $ρ = 0$ , and it is the degenerate end of the theory rather than the general object.

Key theorem with proof Intermediate+

Theorem (The anchor is a Lie-algebra homomorphism). Let $(A \to M, [\cdot, \cdot], ρ)$ satisfy the vector-bundle, antisymmetry, Jacobi, and Leibniz conditions of the definition. Then the anchor on sections $ρ : Γ (A) \to X (M)$ is a homomorphism of Lie algebras: $$ \rho\big([X, Y]\big) = \big[\rho(X), \rho(Y)\big] \qquad \text{for all } X, Y \in \Gamma(A). $$

Proof. Fix $X, Y \in Γ (A)$ , an arbitrary $Z \in Γ (A)$ , and $f \in C^{\infty} (M)$ . The strategy is to compute $[X, [Y, f Z]]$ two ways and compare the coefficients of $Z$ , isolating the function-derivative terms produced by the Leibniz rule ^{[Mackenzie Ch. 3]}.

Apply the Leibniz rule to the inner bracket, then again to the outer one: $$ [Y, fZ] = f[Y, Z] + (\rho(Y)f),Z, $$ $$ \begin{aligned} [X, [Y, fZ]] &= [X,, f[Y,Z]] + [X,, (\rho(Y)f),Z] \ &= f[X,[Y,Z]] + (\rho(X)f)[Y,Z] \ &\quad + (\rho(Y)f)[X,Z] + \big(\rho(X)(\rho(Y)f)\big),Z. \end{aligned} $$

By the Jacobi identity, $[X, [Y, f Z]] = [[X, Y], f Z] + [Y, [X, f Z]]$ . Expand each term by Leibniz: $$ [[X,Y], fZ] = f[[X,Y],Z] + \big(\rho([X,Y])f\big),Z, $$ $$ [Y, [X, fZ]] = f[Y,[X,Z]] + (\rho(X)f)[Y,Z] + (\rho(Y)f)[X,Z] + \big(\rho(Y)(\rho(X)f)\big),Z. $$

Adding the last two displays and using the Jacobi identity once more on the function-coefficient brackets, $f [[X, Y], Z] + f [Y, [X, Z]] = f [X, [Y, Z]]$ , the two expansions of $[X, [Y, f Z]]$ must agree. The terms $f [X, [Y, Z]]$ , $(ρ (X) f) [Y, Z]$ , and $(ρ (Y) f) [X, Z]$ appear identically on both sides and cancel. What remains is the coefficient of $Z$ : $$ \big(\rho(X)(\rho(Y)f)\big),Z = \big(\rho([X,Y])f\big),Z + \big(\rho(Y)(\rho(X)f)\big),Z. $$

Since this holds for every $Z \in Γ (A)$ and the values of sections span each fibre, the function coefficients agree: $$ \rho(X)\big(\rho(Y)f\big) - \rho(Y)\big(\rho(X)f\big) = \rho([X,Y]),f. $$ The left side is by definition the vector-field commutator $[ρ (X), ρ (Y)]$ applied to $f$ . As $f \in C^{\infty} (M)$ was arbitrary and a vector field is determined by its action on functions, $ρ ([X, Y]) = [ρ (X), ρ (Y)]$ . $□$

Bridge. This derivation builds toward the Lie functor that sends each Lie groupoid to its Lie algebroid, where the bracket-homomorphism property of the anchor reappears as the differentiated form of source-fibre composition; it appears again in the action algebroid, where the anchor homomorphism is exactly the statement that an infinitesimal action is a Lie-algebra map into vector fields, and it connects to the cotangent algebroid of a Poisson manifold, where the anchor $π^{♯}$ being a bracket homomorphism encodes the Jacobi identity of the Poisson bracket. Putting these together, the anchor is not free data layered on top of the bracket but a structure the bracket already determines, and recognising this is the first step in classifying algebroids by their anchor image.

Exercises Intermediate+

Exercise 5 (medium, proof).

Show that the anchor image $ρ (A_{x}) \subseteq T_{x} M$ has constant rank along each orbit, where an orbit is a maximal connected integral submanifold of the (possibly singular) distribution $x \mapsto ρ (A_{x})$ . Conclude that on the open set where the rank is locally constant the anchor image integrates to a foliation.

Hint

Use that $ρ$ is a bracket homomorphism so the image is closed under the vector-field bracket; invoke the involutivity criterion of Frobenius 03.02.04.

Answer

The image distribution $D_{x} = ρ (A_{x})$ is spanned by the vector fields $ρ (X)$ , $X \in Γ (A)$ . Since $ρ$ is a bracket homomorphism, $[ρ (X), ρ (Y)] = ρ ([X, Y]) \in D$ , so $D$ is involutive in the sense that brackets of spanning fields stay in $D$ . On the open set where $dim D_{x}$ is locally constant, $D$ is a smooth involutive distribution of fixed rank, hence integrable by the Frobenius theorem 03.02.04, and its integral manifolds foliate that open set. Across rank jumps the orbits still partition $M$ but form a singular foliation.

Exercise 6 (hard, proof).

Let $π$ be a Poisson bivector on $M$ with sharp map $π^{♯} : T^{*} M \to T M$ , $π^{♯} (α) = π (α, \cdot)$ . The cotangent algebroid has $A = T^{*} M$ , anchor $ρ = π^{♯}$ , and bracket determined on exact one-forms by $[df, d g] = d {f, g}$ , where ${f, g} = π (df, d g)$ is the Poisson bracket 05.02.02. Verify the Leibniz rule for this bracket on exact forms, i.e. compute $[df, h d g]$ .

Hint

The Koszul bracket extends the exact-form rule by $[α, h β] = h [α, β] + (π^{♯} (α) h) β$ . Check this is forced.

Answer

The Hamiltonian vector field of $f$ is $X_{f} = π^{♯} (df)$ , and $X_{f} h = {f, h}$ . The cotangent bracket must satisfy $$ [df, h,dg] = h[df, dg] + \big(\pi^\sharp(df)h\big)dg = h,d{f,g} + {f,h},dg. $$ This matches the Leibniz rule with anchor $ρ (df) = π^{♯} (df) = X_{f}$ , since $ρ (df) h = X_{f} h = {f, h}$ . The bracket is the Koszul bracket, and the Jacobi identity for $[\cdot, \cdot]$ is equivalent to the Jacobi identity of the Poisson bracket.

Lean formalization Intermediate+

lean_status: none — Mathlib has LieAlgebra (a module with a bracket satisfying antisymmetry and Jacobi) and an evolving smooth-manifold and vector-bundle library (ContMDiff sections, TangentSpace), but no internalisation of a Lie algebroid: a smooth vector bundle whose section module carries an $R$ -Lie bracket together with an anchor bundle morphism into the tangent bundle, coupled by the Leibniz rule. The intended statement is sketched below in pseudo-Lean to indicate the missing structure; it does not compile against current Mathlib, which is why no Lean module is declared.

-- Pseudo-Lean: target structure, not in Mathlib.
structure LieAlgebroid
    (M : Type*) [SmoothManifold M] (A : VectorBundle M) where
  bracket   : Section A → Section A → Section A
  bracket_antisymm : ∀ X Y, bracket X Y = - bracket Y X
  bracket_jacobi   : ∀ X Y Z,
    bracket X (bracket Y Z) = bracket (bracket X Y) Z + bracket Y (bracket X Z)
  anchor    : BundleMorphism A (TangentBundle M)
  leibniz   : ∀ (X Y : Section A) (f : M → ℝ),
    bracket X (f • Y) = f • bracket X Y + (anchor X • f) • Y
  -- consequence (provable): anchor is a bracket homomorphism

The first genuine obstacle is the section module: Mathlib lacks a packaged "smooth sections of a vector bundle form a $C^{\infty} (M)$ -module supporting a derivation-valued bracket" API, on which the well-typedness of the Leibniz clause depends.

Advanced results Master

The Lie functor. Let $G ⇉ M$ be a Lie groupoid 03.03.10. Its Lie algebroid is the vector bundle $A := ker (d s) ∣_{M}$ , the restriction to the unit submanifold of the kernel of the differential of the source map, with fibre $A_{x} = T_{1_{x}} (s^{- 1} (x))$ the tangent space at the unit to the source fibre through $x$ . The anchor is $ρ := d t ∣_{A} : A \to T M$ , the restriction of the target differential. Right-invariant vector fields on $G$ tangent to the source fibres are in bijection with sections of $A$ , and the bracket of two such right-invariant fields is again right-invariant and source-fibre-tangent, defining the bracket on $Γ (A)$ . The Leibniz rule and the bracket-homomorphism property of $ρ$ are the differentiated forms of the groupoid composition. This is the construction of Pradines ^{[Pradines 1967]}, the groupoid analogue of passing from a Lie group to its Lie algebra by right-invariant fields.

The Atiyah algebroid. Let $P \to M$ be a principal $G$ -bundle 03.05.01. The $G$ -action on $P$ lifts to $T P$ , and the quotient $T P / G \to M$ is a vector bundle of rank $dim M + dim G$ . The differential $d π : T P \to T M$ is $G$ -invariant and descends to an anchor $ρ : T P / G \to T M$ ; the bracket is induced from the Lie bracket of $G$ -invariant vector fields on $P$ , which is closed because the bracket of invariant fields is invariant. The resulting Atiyah algebroid sits in a short exact sequence $$ 0 \longrightarrow \mathrm{ad}(P) \longrightarrow TP/G \xrightarrow{\ \rho\ } TM \longrightarrow 0, $$ where $ad (P) = P \times_{G} g$ is the adjoint bundle, the kernel of the anchor and a bundle of Lie algebras. A splitting of this sequence as vector bundles is precisely a principal connection on $P$ ; the curvature of the connection is the failure of the splitting to be a bracket homomorphism. The Atiyah algebroid is the Lie algebroid of the gauge groupoid $(P \times P) / G$ .

The cotangent algebroid of a Poisson manifold. For a Poisson manifold $(M, π)$ 05.02.02, the cotangent bundle $T^{*} M$ carries a Lie algebroid structure with anchor the sharp map $π^{♯} : T^{*} M \to T M$ and bracket the Koszul bracket, determined on exact forms by $[df, d g] = d {f, g}$ . The anchor image is the characteristic distribution of the Poisson structure, whose integral leaves are the symplectic leaves. That $π^{♯}$ is a bracket homomorphism is equivalent to the Jacobi identity for the Poisson bracket. This identification, due to Weinstein and collaborators ^{[Weinstein 1987]}, makes Poisson geometry a chapter of Lie algebroid theory and is the infinitesimal counterpart of the symplectic groupoid.

Integrability and the failure of Lie's third theorem. Every Lie groupoid differentiates to a Lie algebroid by the Lie functor. The converse fails: not every Lie algebroid is the algebroid of a Lie groupoid. Crainic and Fernandes ^{[Crainic-Fernandes 2003]} identified the precise obstruction. Given a Lie algebroid $A$ , one builds the Weinstein groupoid of $A$ -paths modulo $A$ -homotopy; it is a topological groupoid that carries a smooth structure exactly when two monodromy groups — subgroups of the centres of the isotropy Lie algebras, measured by spherical periods of a transverse symplectic form — are uniformly discrete. When the monodromy groups have arbitrarily small nonzero elements the quotient is not smooth and $A$ does not integrate. No analogous obstruction exists for Lie algebras, where Lie's third theorem holds unconditionally; the monodromy obstruction is genuinely a groupoid phenomenon.

Synthesis. A Lie algebroid $(A \to M, [\cdot, \cdot], ρ)$ is a single bundle-with-bracket-and-anchor that specialises across the whole spectrum of infinitesimal geometry: with $ρ = id$ it is the tangent algebroid $T M$ and the bracket is the vector-field bracket; with $M$ a point it is a Lie algebra; with $ρ = 0$ it is a bundle of Lie algebras with no coupling to the base; with $ρ$ an infinitesimal action it is the action algebroid $g ⋉ M$ ; with $A = T P / G$ it is the Atiyah algebroid whose splittings are connections and whose anchor sequence has adjoint-bundle kernel; and with $A = T^{*} M$ it is the cotangent algebroid of a Poisson manifold whose anchor is the sharp map and whose leaves are symplectic. The anchor is never free data: the Jacobi identity and the Leibniz rule force it to be a bracket homomorphism, so the bracket already knows the anchor, and the rank of the anchor image organises the orbit foliation. Differentiation sends every Lie groupoid to its Lie algebroid, but the monodromy obstruction of Crainic-Fernandes blocks the reverse passage for some algebroids, so the equivalence between global and infinitesimal symmetry that holds for Lie groups breaks for groupoids. Across these constructions the Lie algebroid is the common infinitesimal home for flows, actions, connections, and Poisson structures, and the anchor-and-bracket pair is the minimal data that holds all of them at once.

Full proof set Master

Proposition (The anchor kernel is a bundle of Lie algebras over the regular locus). Let $(A \to M, [\cdot, \cdot], ρ)$ be a Lie algebroid on which $ρ$ has locally constant rank, and set $K_{x} = ker (ρ_{x}) \subseteq A_{x}$ . Then $K = ⨆_{x} K_{x}$ is a smooth subbundle, and the bracket restricts to a $C^{\infty} (M)$ -bilinear, fibrewise Lie-algebra bracket on $Γ (K)$ ; the symbol $⨆$ denotes the disjoint union assembling the kernels into a bundle.

Proof. Where $ρ$ has locally constant rank $r$ , the kernel $K = ker ρ$ is a smooth subbundle of $A$ of rank $rk (A) - r$ , since the kernel of a constant-rank bundle morphism is a subbundle. Take $X, Y \in Γ (K)$ , so $ρ (X) = ρ (Y) = 0$ . By the bracket-homomorphism theorem $ρ ([X, Y]) = [ρ (X), ρ (Y)] = 0$ , so $[X, Y] \in Γ (K)$ and the bracket restricts to $Γ (K)$ . For $C^{\infty} (M)$ -bilinearity on $K$ : by the Leibniz rule $[X, f Y] = f [X, Y] + (ρ (X) f) Y$ , and $ρ (X) = 0$ kills the derivative term, giving $[X, f Y] = f [X, Y]$ ; antisymmetry gives the same in the first slot. A $C^{\infty} (M)$ -bilinear bracket is pointwise, so it restricts to each fibre $K_{x}$ as an antisymmetric bracket, and the Jacobi identity descends fibrewise. Hence each $K_{x}$ is a Lie algebra and $K$ is a bundle of Lie algebras. $□$

Proposition (Anchor sequence of the Atiyah algebroid). Let $P \to M$ be a principal $G$ -bundle and $A = T P / G$ its Atiyah algebroid. Then $0 \to ad (P) \to A ρ T M \to 0$ is an exact sequence of vector bundles, and vector-bundle splittings $σ : T M \to A$ correspond bijectively to principal connections on $P$ .

Proof. The anchor $ρ : T P / G \to T M$ is induced by $d π : T P \to T M$ . Since $π$ is a surjective submersion, $d π$ is fibrewise surjective, so $ρ$ is surjective. The kernel of $d π$ at $p$ is the vertical subspace $V_{p} P ≅ g$ , via the fundamental vector fields of the $G$ -action; quotienting by $G$ identifies $ker ρ$ with the associated bundle $P \times_{G} g = ad (P)$ . Exactness at $A$ follows. A vector-bundle splitting $σ : T M \to A$ with $ρ \circ σ = id$ lifts each tangent vector to a $G$ -invariant horizontal direction, i.e. a $G$ -invariant complement to the vertical bundle in $T P$ ; that is precisely the datum of a principal connection. Conversely a principal connection provides the horizontal lift, hence the splitting. The correspondence is a bijection. $□$

Proposition (Jacobi for the cotangent algebroid equals Poisson Jacobi). Let $(M, π)$ be a manifold with bivector $π$ and Koszul bracket on $Ω^{1} (M)$ determined by $[df, d g] = d {f, g}$ , ${f, g} = π (df, d g)$ , and anchor $π^{♯}$ . The Koszul bracket satisfies the Jacobi identity if and only if ${\cdot, \cdot}$ satisfies the Jacobi identity, i.e. iff $π$ is a Poisson bivector.

Proof. Exact one-forms $df$ generate $Ω^{1} (M)$ as a $C^{\infty} (M)$ -module, and the Leibniz rule extends the bracket from exact generators to all one-forms uniquely. The Koszul-bracket Jacobiator on exact forms is $$ [df, [dg, dh]] + [dg, [dh, df]] + [dh, [df, dg]] = d\Big({f,{g,h}} + {g,{h,f}} + {h,{f,g}}\Big). $$ The right side is the differential of the Poisson Jacobiator of $f, g, h$ . It vanishes for all $f, g, h$ exactly when the Poisson Jacobiator is locally constant in each argument; since it is a derivation in each argument and vanishes when any argument is constant, vanishing of its differential for all inputs is equivalent to its vanishing. By the Leibniz extension the Jacobiator of the Koszul bracket on all one-forms vanishes iff it vanishes on exact forms. Hence the Koszul bracket is a Lie bracket iff ${\cdot, \cdot}$ is, iff $[π, π]_{SN} = 0$ in the Schouten-Nijenhuis bracket. $□$

Connections Master

A Lie groupoid 03.03.10 differentiates to its Lie algebroid by the Lie functor $A = ker (d s) ∣_{M}$ with anchor $d t$ , exactly as a Lie group passes to its Lie algebra. The isotropy Lie groups of the groupoid differentiate to the isotropy Lie algebras that form the anchor kernel, and the source-fibre tangents at the units assemble into the algebroid bundle. This is the channel through which the entire global theory of the previous unit acquires an infinitesimal model.

A Lie algebra 03.04.01 is the special case of a Lie algebroid over a one-point base: the bundle is a single vector space, the anchor lands in the zero tangent space and so vanishes, the Leibniz rule loses its derivative term, and the structure collapses to the antisymmetric Jacobi-satisfying bracket of the Lie-algebra definition. The bundle-of-Lie-algebras case is the fibred generalisation where the same data is spread smoothly over a base with zero anchor.

A principal bundle 03.05.01 yields the Atiyah algebroid $T P / G$ , whose anchor sequence has the adjoint bundle as kernel and whose vector-bundle splittings are exactly principal connections. The curvature of a connection is the failure of its splitting to be a bracket homomorphism, so connection theory is the splitting theory of this one algebroid, and the gauge groupoid of the bundle is the global object the Atiyah algebroid integrates.

The Poisson bracket and Poisson manifold 05.02.02 supply the cotangent algebroid $T^{*} M$ with anchor the sharp map $π^{♯}$ and Koszul bracket on one-forms. The symplectic leaves of the Poisson structure are the orbit foliation of this algebroid, and the equivalence of the Poisson Jacobi identity with the algebroid bracket homomorphism makes Poisson geometry a special chapter of algebroid theory.

Lie's third theorem in its simply-connected form 03.03.06 holds without obstruction for Lie algebras, and its failure for Lie algebroids is the precise content of the Crainic-Fernandes monodromy obstruction. The contrast between the unconditional integrability of Lie algebras and the obstructed integrability of algebroids is the sharpest way the groupoid theory departs from the group theory.

Historical & philosophical context Master

The infinitesimal counterpart of Ehresmann's differentiable groupoids was constructed by Jean Pradines in a sequence of Comptes Rendus notes between 1966 and 1968, where the term Lie algebroid and the differentiation procedure from groupoids to algebroids first appear ^{[Pradines 1967]}. Pradines stated that every differentiable groupoid produces an algebroid and conjectured the converse integrability that was later found to fail. The independent algebraic notion of an anchored bundle with a bracket had appeared earlier in the work of Rinehart on Lie-Rinehart algebras, the module-theoretic shadow of the same structure.

The systematic theory was developed by Kirill Mackenzie, whose 1987 lecture notes and 2005 monograph fixed the standard definition, the Atiyah-sequence picture of connections, and the cohomology theory ^{[Mackenzie 1987; Mackenzie Ch. 3]}. Alan Weinstein and collaborators identified the cotangent algebroid of a Poisson manifold and the symplectic groupoid that integrates it, placing Poisson geometry inside the algebroid framework ^{[Weinstein 1987]}. The integrability question — which algebroids arise from groupoids — was settled by Marius Crainic and Rui Loja Fernandes in 2003, who built the Weinstein groupoid of $A$ -paths and identified the monodromy obstruction in the centres of the isotropy algebras, a phenomenon with no Lie-group analogue.

Bibliography Master

@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}
}

@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}
}

@article{pradines1967,
  author    = {Pradines, Jean},
  title     = {Th\'eorie de Lie pour les groupo\"ides diff\'erentiables. Calcul diff\'erentiel dans la cat\'egorie des groupo\"ides infinit\'esimaux},
  journal   = {Comptes Rendus de l'Acad\'emie des Sciences, Paris, S\'er. A},
  volume    = {264},
  pages     = {245--248},
  year      = {1967}
}

@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}
}

@article{weinstein1987,
  author    = {Weinstein, Alan},
  title     = {Symplectic groupoids and Poisson manifolds},
  journal   = {Bulletin of the American Mathematical Society},
  volume    = {16},
  number    = {1},
  pages     = {101--104},
  year      = {1987}
}

@article{rinehart1963,
  author    = {Rinehart, George S.},
  title     = {Differential forms on general commutative algebras},
  journal   = {Transactions of the American Mathematical Society},
  volume    = {108},
  pages     = {195--222},
  year      = {1963}
}

Lie algebroid — a vector bundle whose sections carry a Lie bracket and an anchor to vector fields obeying the Leibniz law; the infinitesimal counterpart of a Lie groupoid, and the common home of tangent bundles, Lie algebras, action data, Atiyah sequences, and Poisson structures.

Prerequisites

03.03.10
03.05.02
03.04.01
03.02.04

Tier anchors

beginner: the infinitesimal shadow of a system of arrows — at each point, the directions you can move in, bracketing like vector fields
intermediate: Mackenzie — General Theory of Lie Groupoids and Lie Algebroids Ch. 3
master: Mackenzie — General Theory of Lie Groupoids and Lie Algebroids Ch. 3; Crainic-Fernandes — Integrability of Lie brackets (Ann. Math. 157)

References

Mackenzie — General Theory of Lie Groupoids and Lie Algebroids · Ch. 3, the Lie algebroid of a Lie groupoid; anchor and bracket axioms
Mackenzie — Lie Groupoids and Lie Algebroids in Differential Geometry · LMS Lecture Notes 124 (1987), Lie algebroid definition
Crainic-Fernandes — Integrability of Lie brackets · Annals of Mathematics 157 (2003), monodromy obstruction
Pradines — Théorie de Lie pour les groupoïdes différentiables · C. R. Acad. Sci. Paris 264 (1967), infinitesimal calculus of groupoids
Weinstein — Poisson geometry and the cotangent algebroid · Symplectic groupoids and Poisson manifolds, Bull. AMS 16 (1987)

Estimated time

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