Action Lie groupoid and action Lie algebroid

Intuition Beginner

A group action moves points of a space: each group element relocates every point 03.03.02. A Lie groupoid, by contrast, is a collection of reversible arrows between points, where each arrow remembers its start and its end 03.03.10. The action groupoid is the bridge between the two: it turns a moving symmetry into a system of arrows.

The recipe is direct. For a group element $g$ and a starting point $x$, draw one arrow that runs from $x$ to the relocated point $g\cdot x$. Label the arrow by the pair: the element that did the moving and the point it started from. Every group element acting on every point contributes one arrow. So the arrows are exactly the pairs (element, starting point).

Why repackage an action this way? Because the quotient space "points modulo the symmetry" is often badly behaved — points can be glued so tightly that the quotient loses its smoothness. The action groupoid keeps all the moving data in one smooth object instead, so you can still do geometry where the naive quotient fails.

There is a smaller, linear shadow of the same idea. Near the identity of the group, an action becomes a family of velocity fields: each direction in the group pushes the points along a flow. Collecting those velocity fields over the space gives the action algebroid, the infinitesimal version of the action groupoid.

Visual Beginner

Picture a flat space with several marked points, and a circle of rotations acting on it. Pick one point and one rotation. The rotation carries the point to a new location; draw an arrow from the old spot to the new one and tag it with that rotation. Repeat for every rotation and every point.

Two arrows join head-to-tail when the second one ends where the first one starts. Joining "rotate by $g$ starting at $x$" after "rotate by $h$ starting at $g\cdot x$" gives "rotate by $hg$ starting at $x$" — the rotations multiply, the starting point is kept. Each point has a do-nothing arrow tagged by the identity rotation, and reversing an arrow swaps its endpoints and inverts its tag.

Worked example Beginner

Let the group be the four rotations of a square — by $0$, $90$, $180$, $270$ degrees — acting on the four corners labeled $0,1,2,3$. A quarter-turn sends corner $0$ to corner $1$.

The arrow for "quarter-turn, starting at corner $0$" runs from $0$ to $1$. The arrow for "quarter-turn, starting at corner $1$" runs from $1$ to $2$. These two join head-to-tail: start at $0$, quarter-turn to $1$, quarter-turn again to $2$. The combined arrow runs from $0$ to $2$, tagged by the half-turn, because a quarter-turn followed by a quarter-turn is a half-turn.

Count the do-nothing arrows. The identity rotation fixes every corner, so it gives one arrow at each of the four corners that starts and ends in place: four such arrows, one per corner.

Now count arrows that start and end at the same corner. Only the identity rotation keeps a corner where it is (any genuine turn moves all four corners), so each corner has exactly one same-start-same-end arrow. That single arrow per point is the symmetry that fixes that point.

What this tells us: the action's data — which element moves which point where — is faithfully recorded as a system of arrows, and combining arrows mirrors multiplying group elements.

Check your understanding Beginner

Formal definition Intermediate+

Let $G$ be a Lie group 03.03.01 acting smoothly on the left on a manifold $M$ 03.03.02, written $(g,x)\mapsto g\cdot x$. The action groupoid (also translation or transformation groupoid), denoted $G⋉M\rightrightarrows M$, has ^{[Mackenzie Ch. 1 §1.5]}:

• arrow manifold $G\times M$ and object manifold $M$;
• source $s(g,x)=x$ and target $t(g,x)=g\cdot x$, so the arrow $(g,x)$ runs from $x$ to $g\cdot x$;
• composition of $(h,y)$ after $(g,x)$, defined when $y=t(g,x)=g\cdot x$, given by$$(h,g\cdot x)(g,x)=(hg,x);$$
• units $u(x)=(e,x)$, where $e\in G$ is the identity;
• inversion $(g,x{)}^{-1}=(g^{-1},g\cdot x)$.

Here $⋉$ denotes the action-groupoid arrow space $G\times M$ with the structure above; the symbol $\rightrightarrows$ is the standard shorthand for the source–target pair $(s,t)$ from arrows to objects. The composition is well posed: the source of $(hg,x)$ is $x=s(g,x)$ and its target is $(hg)\cdot x=h\cdot (g\cdot x)=t(h,g\cdot x)$, using the action axiom $(hg)\cdot x=h\cdot (g\cdot x)$. Associativity and the unit and inverse laws reduce to the group laws of $G$ and the action axioms; the verification is carried out in the key theorem below.

For each $x\in M$ the isotropy group is $$ (G \ltimes M)_x^x = {(g, x) : g \cdot x = x} ;\cong; G_x, $$ the stabilizer subgroup of $x$ under the action 03.03.02. The orbit of $x$ in the groupoid sense — the set of objects reachable from $x$ by an arrow — is $\{g\cdot x:g\in G\}=G\cdot x$, the action orbit. The groupoid is transitive exactly when there is a single orbit, that is, when the action is transitive.

A non-example marks the boundary of the construction. If the "action" fails the axiom $(hg)\cdot x=h\cdot (g\cdot x)$ — for instance an assignment $(g,x)\mapsto g\cdot x$ that is not associative over $G$ — then the composition $(hg,x)$ no longer matches $(h,g\cdot x)(g,x)$ as endpoints, and $G\times M$ carries no groupoid structure of this form. The associativity of the action is precisely what makes the partial multiplication associative.

The action (Lie) algebroid of the same action, denoted $\mathfrak{g}⋉M\to M$, is the infinitesimal counterpart ^{[Mackenzie Ch. 3 §1.4]}. It is the product vector bundle $M\times \mathfrak{g}\to M$ over $M$ with fibre the Lie algebra $\mathfrak{g}$ of $G$, equipped with:

• the anchor $\rho :M\times \mathfrak{g}\to TM$, $\rho (x,\xi )={\xi }_M(x)$, where ${\xi }_M$ is the fundamental vector field (infinitesimal generator) of $\xi \in \mathfrak{g}$, defined by$${\xi }_M(x)={\frac{d}{dt}|}_{t=0}\exp (t\xi )\cdot x;$$
• the bracket on sections $\Gamma (M\times \mathfrak{g})=C^{\infty }(M,\mathfrak{g})$ that extends the Lie bracket of $\mathfrak{g}$ on constant sections by the Leibniz rule: for constant sections $\xi ,\eta \in \mathfrak{g}$,$$[\xi ,\eta {]}_{\mathfrak{g}⋉M}=[\xi ,\eta {]}_{\mathfrak{g}},$$ and for $f,h\in C^{\infty }(M)$,$$[f\xi ,h\eta ]=fh[\xi ,\eta {]}_{\mathfrak{g}}+f(\rho (\xi )h)\eta -h(\rho (\eta )f)\xi .$$

The Leibniz rule $[X,hY]=h[X,Y]+(\rho (X)h)Y$ for the algebroid bracket is the defining compatibility of any Lie algebroid 03.04.16; here it forces the formula above once the bracket is fixed on constants. The symbol $\rho (\xi )h$ means the derivative of $h$ along the vector field $\rho (x,\xi )={\xi }_M(x)$.

Key theorem with proof Intermediate+

Theorem (The action groupoid is a Lie groupoid, with stabilizer isotropy). Let a Lie group $G$ act smoothly on a manifold $M$. Then $G⋉M\rightrightarrows M$ with the source, target, composition, units, and inversion above is a Lie groupoid. Its isotropy group at $x$ is the stabilizer $G_x$, and its orbits are the action orbits.

Proof. The arrow space $G\times M$ and the object space $M$ are smooth manifolds 03.03.10, the former of dimension $\dim G+\dim M$.

Source is a surjective submersion. The source $s(g,x)=x$ is the projection $G\times M\to M$, which is a surjective submersion.

Target is a surjective submersion. Write $a:G\times M\to M$ for the smooth action map, so $t=a$. Fix $(g,x)$. The partial map $x\mapsto g\cdot x$ is a diffeomorphism of $M$ with inverse $y\mapsto g^{-1}\cdot y$, because $(g^{-1}g)\cdot x=e\cdot x=x$. Hence the restriction of $t$ to $G\times \{x{\}}^c$ — more precisely, to the slice $\{g\}\times M$ through $(g,x)$ — is already a diffeomorphism onto $M$, so $d{t}_{(g,x)}$ is surjective. Thus $t$ is a submersion at every point, and it is surjective because $t(e,y)=y$ for all $y$.

Composable pairs and smooth multiplication. A pair $((h,y),(g,x))$ is composable when $s(h,y)=t(g,x)$, i.e. $y=g\cdot x$. The set of composable pairs is therefore $$ {\big((h, g\cdot x), (g, x)\big) : h, g \in G,\ x \in M}, $$ which is the graph of the smooth map $(h,g,x)\mapsto ((h,g\cdot x),(g,x))$ and so is a closed embedded submanifold of $(G\times M{)}^2$ diffeomorphic to $G\times G\times M$. Under this identification the multiplication is $$ m\big((h, g\cdot x), (g, x)\big) = (hg,, x), $$ which is smooth because $(h,g,x)\mapsto (hg,x)$ is smooth (group multiplication is smooth in $G$).

Associativity. For composable $(k,z),(h,y),(g,x)$ with $y=g\cdot x$ and $z=h\cdot y=(hg)\cdot x$, $$ \big((k, z)(h, y)\big)(g, x) = (kh,, y)(g, x) = \big((kh)g,, x\big), $$ $$ (k, z)\big((h, y)(g, x)\big) = (k, z)(hg,, x) = \big(k(hg),, x\big), $$ and these agree by associativity of $G$.

Units. The unit map $u(x)=(e,x)$ is a smooth embedding with $s(e,x)=x=t(e,x)$. For any $(g,x)$, $$ (e,, g \cdot x)(g, x) = (eg,, x) = (g, x), \qquad (g, x)(e, x) = (ge,, x) = (g, x), $$ so $u(t(g,x))$ and $u(s(g,x))$ act as left and right identities.

Inversion. Set $i(g,x)=(g^{-1},g\cdot x)$, a smooth involution since $i(i(g,x))=i(g^{-1},g\cdot x)=(g,g^{-1}\cdot (g\cdot x))=(g,x)$. It interchanges source and target: $s(g^{-1},g\cdot x)=g\cdot x=t(g,x)$ and $t(g^{-1},g\cdot x)=g^{-1}\cdot (g\cdot x)=x=s(g,x)$. Moreover $$ (g, x)^{-1}(g, x) = (g^{-1}, g\cdot x)(g, x) = (g^{-1} g,, x) = (e, x) = u(s(g,x)), $$ and symmetrically $(g,x)(g,x{)}^{-1}=(e,g\cdot x)=u(t(g,x))$.

All defining clauses of a Lie groupoid hold, so $G⋉M$ is a Lie groupoid.

Isotropy and orbits. The isotropy at $x$ is the set of arrows with $s=t=x$, i.e. $(g,x)$ with $g\cdot x=x$; the assignment $(g,x)\mapsto g$ is a Lie-group isomorphism onto $G_x=\{g:g\cdot x=x\}$. The orbit of $x$ is $t(s^{-1}(x))=\{g\cdot x:g\in G\}=G\cdot x$, the action orbit. $\square$

Bridge. This construction builds toward the infinitesimal theory of the same action, where differentiating the source fibres at the units produces the action algebroid treated in the Advanced results; it appears again in the gauge-groupoid picture 03.05.21, since a transitive action makes $G⋉M$ the gauge groupoid of the bundle $G\to G/G_x$. It also reappears whenever a quotient $M/G$ is singular, because the action groupoid is the smooth stand-in for the bad quotient, and it organises the dictionary in which orbits, stabilizers, and transitivity of the action become orbits, isotropy, and transitivity of the groupoid 03.03.10. Reading the isotropy off the action is the first step in passing from the global groupoid to its linear model.

Exercises Intermediate+

Lean formalization Intermediate+

lean_status: none — Mathlib has the algebraic action hierarchy (MulAction, SMul) and an evolving smooth-manifold library, but no LieGroupoid object and no LieAlgebroid object, so neither the action groupoid nor the action algebroid can be stated against current Mathlib. The intended structure is sketched below in pseudo-Lean; it does not compile, which is why no Lean module is declared.

-- Pseudo-Lean: target structures, not in Mathlib.
variable (G : Type*) [LieGroup G] (M : Type*) [Manifold M] [SmoothMulAction G M]

def actionGroupoid : LieGroupoid M (G × M) where
  source := fun p => p.2                  -- s(g, x) = x
  target := fun p => p.1 • p.2            -- t(g, x) = g • x
  mul    := fun h g => (h.1 * g.1, g.2)   -- (h, g•x)(g, x) = (hg, x)
  unit   := fun x => (1, x)               -- u(x) = (e, x)
  inv    := fun p => (p.1⁻¹, p.1 • p.2)   -- (g, x)⁻¹ = (g⁻¹, g•x)
  -- source_submersion, target_submersion, assoc, unit, inv axioms ...

def actionAlgebroid : LieAlgebroid M (M × LieAlgebra G) where
  anchor := fun x ξ => fundamentalVF ξ x  -- ρ(x, ξ) = ξ_M(x)
  bracket := extendByLeibniz (lieBracket : LieAlgebra G → LieAlgebra G → _)
  -- leibniz, jacobi axioms ...

The first genuine obstacle is fundamentalVF: there is no Mathlib map sending $\xi \in \mathfrak{g}$ to the vector field $x\mapsto \frac{d}{dt}{|}_0\exp (t\xi )\cdot x$, nor the theorem that $\xi \mapsto {\xi }_M$ is a Lie-algebra (anti)homomorphism, on which the bracket morphism property of the anchor depends.

Advanced results Master

The action algebroid is the Lie algebroid of the action groupoid. Applying the Lie functor to $G⋉M$ — restricting the kernel of $ds$ to the units and taking $dt$ as anchor 03.04.16 — returns the product bundle $M\times \mathfrak{g}$ with anchor $\rho (x,\xi )={\xi }_M(x)$ and the Leibniz bracket extending the bracket of $\mathfrak{g}$ on constant sections. This is the infinitesimal counterpart of the groupoid, and it integrates back to $G⋉M$ whenever $G$ is connected and simply connected; for general $G$ the source-simply-connected integration is $\tilde{G}⋉M$ for the universal cover $\tilde{G}$.

The anchor as an (anti)homomorphism. For a left action with the convention $(hg)\cdot x=h\cdot (g\cdot x)$, the infinitesimal-generator map $\xi \mapsto {\xi }_M$ satisfies $$ [\xi, \eta]_M = -[\xi_M, \eta_M], $$ so it is a Lie-algebra antihomomorphism into the vector fields under the standard bracket of vector fields. Replacing the left action by the corresponding right action $x\cdot g:=g^{-1}\cdot x$, or flipping the sign convention on $\mathfrak{g}$, turns it into a homomorphism. The action algebroid bracket is arranged so that its anchor $\rho$ is in either case a bracket morphism onto the image vector fields, because the algebroid bracket already absorbs the sign through the placement of the action on the correct side. This is the precise content of the equivalence: $\rho$ is a bracket morphism on $\mathfrak{g}⋉M$ if and only if $\xi \mapsto {\xi }_M$ is a Lie-algebra (anti)homomorphism with the matching convention.

Isotropy algebroid and the kernel of the anchor. The kernel of $\rho$ over a point $x$ is $\{(x,\xi ):{\xi }_M(x)=0\}=\{x\}\times {\mathfrak{g}}_x$, where ${\mathfrak{g}}_x=\{\xi :{\xi }_M(x)=0\}$ is the Lie algebra of the stabilizer $G_x$ — the isotropy Lie algebra at $x$. Where the action has locally constant orbit dimension, these assemble into the isotropy bundle of Lie algebras, the infinitesimal version of the isotropy groups $G_x$ of the groupoid. The image of $\rho$ is the tangent distribution to the orbits; on the open dense set where the orbit dimension is locally constant, that image is an involutive distribution integrating to the orbit foliation.

Proper and free actions, and the quotient. If the action is free then every stabilizer is the one-element group, so $G⋉M$ has one-element isotropy; if it is also proper, the orbit space $M/G$ is a smooth manifold and $G⋉M$ is isomorphic to the pullback of the unit groupoid $M/G\rightrightarrows M/G$ along $M\to M/G$ — equivalently, $G⋉M$ is Morita equivalent to the manifold $M/G$ viewed as the unit groupoid. When the action is proper but not free, $M/G$ is an orbifold and the action groupoid presents the quotient stack $[M/G]$, the object that remembers the stabilizers the naive quotient discards.

Transitive case and the gauge groupoid. A transitive action makes $G⋉M$ a transitive Lie groupoid, hence the gauge groupoid $(G\times G)/G_{x_0}$ of the principal bundle $G\to G/G_{x_0}\cong M$ 03.05.21. On the algebroid side this is the statement that $\mathfrak{g}⋉(G/H)$ is the Atiyah algebroid of $G\to G/H$, surjective anchor, with kernel the adjoint bundle $G{\times }_H\mathfrak{h}$.

Synthesis. The action groupoid $G⋉M$ records a smooth action as a single Lie groupoid whose source is a projection, whose target is the action map, and whose isotropy at each point is the stabilizer $G_x$, so that the entire orbit–stabilizer dictionary of group actions becomes the orbit–isotropy dictionary of groupoids; transitivity, freeness, and properness of the action read off as transitivity of the groupoid, one-element isotropy, and smoothness of the quotient stack respectively. Differentiating $G⋉M$ by the Lie functor produces the action algebroid $\mathfrak{g}⋉M=M\times \mathfrak{g}$, the product bundle whose anchor $\rho (x,\xi )={\xi }_M(x)$ sends each Lie-algebra element to its fundamental vector field and whose bracket extends the bracket of $\mathfrak{g}$ by the Leibniz rule, with the anchor a bracket morphism exactly when $\xi \mapsto {\xi }_M$ is a Lie-algebra (anti)homomorphism for the matching action convention. The kernel of $\rho$ is the bundle of isotropy Lie algebras ${\mathfrak{g}}_x$, the image of $\rho$ is the tangent distribution to the orbits, and in the transitive case the groupoid is the gauge groupoid of $G\to G/G_{x_0}$ and the algebroid is the corresponding Atiyah algebroid. Across the construction the action groupoid and action algebroid are the global and infinitesimal homes of a symmetry that moves points, providing a smooth replacement for the quotient $M/G$ precisely where that quotient is singular, and supplying the canonical example on which the groupoid–algebroid correspondence is calibrated.

Full proof set Master

Proposition (Composable arrows form a manifold, multiplication is smooth). For a smooth action of $G$ on $M$, the set of composable pairs of $G⋉M$ is a closed embedded submanifold of $(G\times M{)}^2$ diffeomorphic to $G\times G\times M$, and the partial multiplication is smooth.

Proof. A pair $((h,y),(g,x))$ is composable iff $s(h,y)=t(g,x)$, i.e. $y=g\cdot x$. The map $$ \Theta : G \times G \times M \to (G \times M)^2, \qquad (h, g, x) \mapsto \big((h,, g \cdot x),, (g, x)\big) $$ is a smooth injective immersion: its first component records $(h,g\cdot x)$ and its second records $(g,x)$, and the projection $((h,y),(g,x))\mapsto (h,g,x)$ recovers $(h,g,x)$ smoothly from the image since $g$ and $x$ are read off the second factor and $h$ off the first. Hence $\Theta$ is a smooth embedding with closed image equal to the composable set $G{\times }_{M}G$. Under $\Theta$ the multiplication becomes $(h,g,x)\mapsto (hg,x)$, smooth because multiplication $G\times G\to G$ is smooth. So $m:G{\times }_{M}G\to G\times M$ is smooth. $\square$

Proposition (The infinitesimal-generator map is a Lie-algebra antihomomorphism). For a smooth left action of $G$ on $M$ with $(hg)\cdot x=h\cdot (g\cdot x)$, the map $\mathfrak{g}\to \mathfrak{X}(M)$, $\xi \mapsto {\xi }_M$, satisfies $[\xi ,\eta {]}_M=-[{\xi }_M,{\eta }_M]$ for all $\xi ,\eta \in \mathfrak{g}$.

Proof. For $g\in G$ let $a_g:M\to M$, $a_g(x)=g\cdot x$, a diffeomorphism with $a_{hg}=a_h\circ a_g$. The fundamental field is ${\xi }_M(x)=\frac{d}{dt}{|}_0{a}_{\exp (t\xi )}(x)$. The flow of ${\xi }_M$ is ${\phi }_t^{{\xi }_M}=a_{\exp (t\xi )}$, since $\frac{d}{dt}{a}_{\exp (t\xi )}(x)={\xi }_M(a_{\exp (t\xi )}(x))$ by the one-parameter-subgroup property $\exp ((t+s)\xi )=\exp (t\xi )\exp (s\xi )$ and $a_{\exp (t\xi )\exp (s\xi )}=a_{\exp (t\xi )}\circ a_{\exp (s\xi )}$. The Lie bracket of vector fields is the derivative of the pushforward along the flow: $$ [\xi_M, \eta_M] = \left.\frac{d}{dt}\right|0 \big(\varphi^{\xi_M}{-t}\big)* \eta_M = \left.\frac{d}{dt}\right|0 (a{\exp(-t\xi)})* \eta_M. $$ Now $(a_g{)}_{\ast }{\eta }_M$ is the fundamental field of ${\mathrm{Ad}}_g\eta$: differentiating $a_g(a_{\exp (s\eta )}(x))=a_{g\exp (s\eta )}(x)=a_{g\exp (s\eta )g^{-1}}(a_g(x))$ at $s=0$ gives $(a_g{)}_{\ast }{\eta }_M(a_g(x))=({\mathrm{Ad}}_g\eta {)}_M(a_g(x))$, hence $(a_g{)}_{\ast }{\eta }_M=({\mathrm{Ad}}_g\eta {)}_M$. Therefore $$ [\xi_M, \eta_M] = \left.\frac{d}{dt}\right|0 (\mathrm{Ad}{\exp(-t\xi)} \eta)_M = \Big(\left.\tfrac{d}{dt}\right|0 \mathrm{Ad}{\exp(-t\xi)} \eta\Big)M = (-\mathrm{ad}\xi \eta)_M = (-[\xi, \eta])M, $$ using $\frac{d}{dt}\big|0 \mathrm{Ad}{\exp(t\xi)} = \mathrm{ad}\xi$.Thus$[\xi_M, \eta_M] = -[\xi, \eta]_M$,i.e.$[\xi, \eta]_M = -[\xi_M, \eta_M]$.$\square$

Proposition (The anchor is a bracket morphism). On the action algebroid $\mathfrak{g}⋉M$, the anchor $\rho$ satisfies $\rho ([X,Y])=[\rho (X),\rho (Y)]$ for all sections $X,Y\in C^{\infty }(M,\mathfrak{g})$.

Proof. It suffices to check on constant sections $\xi ,\eta \in \mathfrak{g}$ and extend by the Leibniz rule, since both sides are first-order and agree on the $C^{\infty }(M)$-module generators. On constants, the algebroid bracket is defined so that $\rho ([\xi ,\eta {]}_{\mathfrak{g}⋉M})$ matches $[\rho (\xi ),\rho (\eta )]$. With the left-action convention the algebroid bracket on constants is taken as $[\xi ,\eta {]}_{\mathfrak{g}⋉M}=-[\xi ,\eta {]}_{\mathfrak{g}}$ (absorbing the antihomomorphism sign), so that $$ \rho([\xi, \eta]{\mathfrak g \ltimes M}) = (-[\xi, \eta]{\mathfrak g})M = -(-[\xi_M, \eta_M]) = [\xi_M, \eta_M] = [\rho(\xi), \rho(\eta)], $$ by the previous proposition. For general sections $X=f^i{\xi }_i$, $Y=h^j{\eta }_j$ the Leibniz rule propagates the identity: $$ \rho([f^i \xi_i, h^j \eta_j]) = \rho\big(f^i h^j [\xi_i, \eta_j] + f^i(\rho(\xi_i) h^j)\eta_j - h^j(\rho(\eta_j) f^i)\xi_i\big), $$ and applying $\rho$ — which is $C^{\infty }(M)$-linear — together with the constant-section identity and the product rule for $\rho(\xi_i) h^j = \xi{i,M}(h^j)$reproduces$[\rho(X), \rho(Y)] = [f^i \xi_{i,M}, h^j \eta_{j,M}]$expandedbytheLeibnizruleforthebracketofvectorfields.Hence$\rho$isabracketmorphism.$\square$

Connections Master

A group action 03.03.02 is the input data of this unit, and the dictionary is exact: orbits of the action are groupoid orbits, stabilizers $G_x$ are the isotropy groups, freeness is one-element isotropy, and transitivity of the action is transitivity of the groupoid. The action groupoid is what lets a quotient $M/G$ that is singular as a space be replaced by a smooth groupoid carrying the same orbit data.

A Lie group 03.03.01 is the one-object special case: when $M$ is a point, the only action is the fixed-point action, $G⋉\{\ast \}=G\rightrightarrows \{\ast \}$ is the Lie group regarded as a one-object Lie groupoid, and its action algebroid $\mathfrak{g}⋉\{\ast \}=\mathfrak{g}$ is the Lie algebra over a point. The construction therefore interpolates between a bare Lie group and a smoothly varying family of symmetries.

A Lie groupoid 03.03.10 receives $G⋉M$ as one of its four canonical examples, alongside the pair groupoid, the fundamental groupoid, and the gauge groupoid; the action groupoid is the example on which the isotropy theorem and the orbit description of the general theory are most concretely visible, since both reduce to stabilizers and action orbits.

The gauge groupoid 03.05.21 is the transitive face of this unit: a transitive action makes $G⋉M$ the gauge groupoid of $G\to G/G_{x_0}$, and conversely every transitive action groupoid is a gauge groupoid. This is the action-theoretic instance of Mackenzie's correspondence between transitive Lie groupoids and principal bundles.

The Lie algebroid 03.04.16 is the target of the differentiation in this unit: $\mathfrak{g}⋉M$ is the action algebroid, the simplest non-tangent, non-bundle-of-Lie-algebras example, and it is the test case on which the anchor, the Leibniz bracket, and the integrability question are calibrated.

Historical & philosophical context Master

The differentiable groupoid was introduced by Charles Ehresmann in 1959 within his program on differentiable categories and fibre bundles, and the action (translation) groupoid appears there as a basic construction associated to a Lie group acting on a manifold ^{[Ehresmann 1959]}. The infinitesimal counterpart was developed by Jean Pradines in his 1966–1968 notes, where the differentiation of a differentiable groupoid to its algebroid was first stated; the action algebroid is the linearisation of Ehresmann's action groupoid under that procedure ^{[Pradines 1966]}.

Kirill Mackenzie's 1987 lecture notes and 2005 monograph fixed the now-standard treatment, presenting the action groupoid in Chapter 1 and the action (transformation) Lie algebroid in Chapter 3, including the equivalence between the bracket-morphism property of the anchor and the (anti)homomorphism property of the infinitesimal-generator map ^{[Mackenzie 1987; Mackenzie Ch. 1 §1.5]}. Ieke Moerdijk and Janez Mrcun's 2003 text uses the translation groupoid as the organising example linking group actions, foliations, and orbit spaces, and develops the Morita-equivalence viewpoint under which a free proper action groupoid presents its smooth quotient ^{[Moerdijk-Mrcun 2003]}. The integrability question for the action algebroid — when $\mathfrak{g}⋉M$ integrates to a Lie groupoid — was settled in general by Marius Crainic and Rui Loja Fernandes in 2003, who showed action algebroids are always integrable, in contrast to the general failure of Lie's third theorem for algebroids.

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}
}

@book{moerdijk-mrcun2003,
  author    = {Moerdijk, Ieke and Mr\v{c}un, Janez},
  title     = {Introduction to Foliations and Lie Groupoids},
  series    = {Cambridge Studies in Advanced Mathematics},
  volume    = {91},
  publisher = {Cambridge University Press},
  year      = {2003}
}

@incollection{ehresmann1959,
  author    = {Ehresmann, Charles},
  title     = {Cat\'egories topologiques et cat\'egories diff\'erentiables},
  booktitle = {Colloque de G\'eom\'etrie Diff\'erentielle Globale (Bruxelles, 1958)},
  pages     = {137--150},
  publisher = {Centre Belge de Recherches Math\'ematiques},
  year      = {1959}
}

@article{pradines1966,
  author    = {Pradines, Jean},
  title     = {Th\'eorie de Lie pour les groupo\"ides diff\'erentiables. Relations entre propri\'et\'es locales et globales},
  journal   = {Comptes Rendus de l'Acad\'emie des Sciences, Paris},
  volume    = {263},
  pages     = {907--910},
  year      = {1966}
}

@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}
}

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Action Lie groupoid — a smooth action $G\times M\to M$ repackaged as the groupoid $G⋉M\rightrightarrows M$ with arrows $(g,x):x\to g\cdot x$, isotropy the stabilizers, orbits the action orbits; its linearisation is the action Lie algebroid $\mathfrak{g}⋉M$ with anchor the fundamental-vector-field map.