Atiyah algebroid of a principal bundle

Intuition Beginner

A principal bundle spreads a copy of a symmetry group over every point of a base space. The gauge groupoid collects every allowed comparison between fibres into reversible arrows. Now zoom all the way in: instead of finite comparisons, ask about the infinitesimal ones, the directions you can nudge the whole structure in. That infinitesimal object is the Atiyah algebroid.

Here is the honest picture. A direction you can move the total space in either pushes the base point somewhere, or stays over the same base point and only twists inside the fibre, or does some of both. The Atiyah algebroid is the bundle of all these allowed infinitesimal moves, with one extra rule: you only keep the ones that respect the spread-out symmetry. Such a move is the same at every point of a fibre.

Why bother? Because a connection, the rule that says how to carry data along the base without spurious twisting, is exactly a choice of "for each base direction, one preferred invariant move that pushes the base point that way and adds no extra twist." Curvature then measures how badly those preferred moves fail to combine cleanly.

So the slogan is short. The gauge groupoid is every finite comparison of fibres; the Atiyah algebroid is every invariant infinitesimal move, and a connection is a clean way to lift base directions into it.

Visual Beginner

Picture the base as a curved surface with one marked point. Above it floats the fibre, a copy of the symmetry group drawn as a small disc. An invariant infinitesimal move is an arrow attached to the total space that looks the same over the whole fibre: part of it points along the base surface, part of it spins inside the disc.

The downward shadow of the arrow on the base is its anchor image: the base direction it makes you move in. When the shadow is zero, the move is pure fibre twist and changes nothing on the base; these pure twists form a smaller bundle sitting inside, the adjoint bundle. A connection picks, for each base direction, one shadow-matching move with no leftover twist.

Worked example Beginner

Take the simplest case: the base is a flat disc, and over every point we glue the same circle of rotation angles. The total space is the product of the disc with the circle, and a point is a base point together with a chosen angle. This is a product bundle.

What are the invariant infinitesimal moves? A move can push the base point in some flat direction on the disc, and it can add a fixed turn rate to the angle. Because everything is a product, "fixed turn rate" already means the same thing over the whole fibre, so every such move is automatically invariant. So an invariant move is a base direction together with one number, the turn rate.

Count the pieces. The base directions over a point form a two-dimensional flat space. The turn rates form a one-dimensional line. Together an invariant move is described by three numbers: two for the base push and one for the twist. The pure twists, the moves with zero base push, are the single turn-rate line.

What this tells us: for a product bundle the Atiyah algebroid splits cleanly into base directions plus fibre twists, with no mixing forced on you. The split itself, "push the base and add zero twist," is the flat connection. Every other Atiyah algebroid is built to imitate this product case locally, but the clean split need not extend globally, and that failure is what connections and curvature track.

Check your understanding Beginner

Formal definition Intermediate+

Let $\pi :P\to M$ be a principal $G$-bundle 03.05.01, with $G$ a Lie group acting on $P$ on the right, smoothly, freely, and properly 03.05.07. The derivative of the right action lifts $G$ to act on the tangent bundle $TP$: for $h\in G$ write $R_h:P\to P$ for the action of $h$, and let $G$ act on $TP$ by $h\cdot v:=(d{R}_h)v$ for $v\in TP$. This lifted action is again free and proper, so the quotient $$ \mathrm{At}(P) := \frac{TP}{G} $$ is a smooth vector bundle over $M=P/G$ ^{[Mackenzie Ch. III §3]}. Its rank is $\dim M+\dim G$.

Definition (Atiyah algebroid). The Atiyah algebroid of $P$ is the vector bundle $\mathrm{At}(P)=TP/G\to M$ equipped with:

• the anchor ${\pi }_{\ast }:TP/G\to TM$, the vector-bundle morphism descending the differential $d\pi :TP\to TM$, which is $G$-invariant because $\pi \circ R_h=\pi$;
• the bracket $[\cdot ,\cdot ]$ on $\Gamma (\mathrm{At}(P))$ obtained by identifying sections of $TP/G$ with $G$-invariant vector fields on $P$ and taking the Lie bracket of vector fields 03.04.01.

The identification of sections is the structural fact. A section of $TP/G$ over $M$ is the same datum as a $G$-invariant vector field on $P$: a vector field $X\in \mathfrak{X}(P)$ with $(d{R}_h)X=X\circ R_h$ for every $h\in G$. Invariant fields descend to sections of the quotient, and every section lifts uniquely to an invariant field, so $\Gamma (\mathrm{At}(P))\cong \mathfrak{X}(P{)}^G$ as $C^{\infty }(M)$-modules, with $C^{\infty }(M)=C^{\infty }(P{)}^G$ acting by pullback. Under this identification the anchor sends an invariant field $X$ to its projection ${\pi }_{\ast }X\in \mathfrak{X}(M)$, well defined because $X$ is $\pi$-related to a single field downstairs.

The kernel of the anchor is the adjoint bundle $$ \mathrm{ad}(P) := \frac{P \times \mathfrak g}{G}, \qquad h \cdot (p, \xi) = (p \cdot h, \ \mathrm{Ad}{h^{-1}} \xi), $$ where $\mathfrak{g}=\mathrm{Lie}(G)$. Its sections are the $G$-invariant vertical vector fields on $P$, those tangent to the fibres. The notation: $\mathfrak{X}(P{)}^G$ denotes the $G$-invariant vector fields on $P$, $\mathrm{Ad}$ the adjoint representation of $G$ on $\mathfrak{g}$, and $\zeta\xi$belowthefundamentalvectorfieldof$\xi \in \mathfrak g$,namely$\zeta_\xi(p) = \tfrac{d}{dt}\big|_0, p \cdot \exp(t\xi)$.

A non-example marks the boundary. If one drops freeness of the $G$-action, the lifted action on $TP$ need not be free, the quotient $TP/G$ need not be a vector bundle, and the anchor sequence below need not be exact. Freeness and properness of the principal action are exactly what place $\mathrm{At}(P)$ in the category of smooth vector bundles and make ${\pi }_{\ast }$ a constant-rank surjection.

Key theorem with proof Intermediate+

Theorem (the Atiyah sequence is a short exact sequence of vector bundles, and $\mathrm{At}(P)$ is a Lie algebroid). Let $\pi :P\to M$ be a principal $G$-bundle. The sequence $$ 0 \longrightarrow \mathrm{ad}(P) \xrightarrow{\ \iota\ } \mathrm{At}(P) \xrightarrow{\ \pi_\ast\ } TM \longrightarrow 0 $$ of vector bundles over $M$ is exact. The bracket of $G$-invariant vector fields is closed on $\mathfrak{X}(P{)}^G$, and $(\mathrm{At}(P),[\cdot ,\cdot ],{\pi }_{\ast })$ is a Lie algebroid with anchor ${\pi }_{\ast }$. ^{[Mackenzie Ch. III §3]}

Proof. Write $V:=\ker (d\pi )\subseteq TP$ for the vertical bundle, the tangent spaces to the fibres.

The vertical bundle is $\mathrm{ad}(P)$ after quotient. The fundamental-vector-field map $P\times \mathfrak{g}\to TP$, $(p,\xi )\mapsto {\zeta }_{\xi }(p)$, is fibrewise injective because the action is free, and its image is exactly $V$: at each $p$ the map $\xi \mapsto {\zeta }_{\xi }(p)$ is a linear isomorphism $\mathfrak{g}\stackrel{\sim }{\to }{V}_p$, since both have dimension $\dim G$ and the map is injective. This map intertwines the $G$-action $h\cdot (p,\xi )=(p\cdot h,{\mathrm{Ad}}_{h^{-1}}\xi )$ on the source with the lifted action on $TP$: differentiating $R_h(p\cdot \exp (t\xi ))=(p\cdot h)\cdot \exp (t{\mathrm{Ad}}_{h^{-1}}\xi )$ at $t=0$ gives $(d{R}_h){\zeta }_{\xi }(p)={\zeta }_{{\mathrm{Ad}}_{h^{-1}}\xi }(p\cdot h)$. Quotienting by $G$ yields a vector-bundle isomorphism $\iota :\mathrm{ad}(P)=(P\times \mathfrak{g})/G\stackrel{\sim }{\to }V/G\hookrightarrow TP/G=\mathrm{At}(P)$.

Exactness. The differential $d\pi :TP\to {\pi }^{\ast }TM$ is $G$-invariant and fibrewise surjective with kernel $V$, of constant rank $\dim M$. Being a surjective submersion in each entry, $\pi$ makes $d\pi$ a surjection of vector bundles, so the descended anchor ${\pi }_{\ast }:TP/G\to TM$ is surjective. Its kernel is $V/G=\mathrm{im}(\iota )$ by the previous paragraph. Injectivity of $\iota$ is inherited from the fibrewise injectivity of the fundamental-field map. Hence the sequence is exact at all three positions.

Closure of the bracket. Let $X,Y\in \mathfrak{X}(P{)}^G$ be invariant. For each $h$, the pushforward $(R_h{)}_{\ast }$ is a diffeomorphism, and the Lie bracket is natural under diffeomorphisms: $(R_h{)}_{\ast }[X,Y]=[(R_h{)}_{\ast }X,(R_h{)}_{\ast }Y]$. Invariance of $X$ and $Y$ means $(R_h{)}_{\ast }X=X$ and $(R_h{)}_{\ast }Y=Y$, so $(R_h{)}_{\ast }[X,Y]=[X,Y]$, i.e. $[X,Y]$ is invariant. Thus $\mathfrak{X}(P{)}^G$ is closed under the bracket, which descends to a bracket on $\Gamma (\mathrm{At}(P))$.

Lie algebroid axioms. The descended bracket is $\mathbb{R}$-bilinear, antisymmetric, and satisfies the Jacobi identity because the vector-field bracket on $P$ does. The anchor ${\pi }_{\ast }$ is a vector-bundle morphism over ${\mathrm{id}}_M$. For the Leibniz rule, let $X,Y\in \mathfrak{X}(P{)}^G$ and $f\in C^{\infty }(M)=C^{\infty }(P{)}^G$, viewing $f$ as an invariant function on $P$. Then $$ [X, fY] = f[X, Y] + (Xf), Y . $$ Since $X$ is $\pi$-related to ${\pi }_{\ast }X$, the derivative $Xf$ of the basic function $f=g\circ \pi$ equals $(({\pi }_{\ast }X)g)\circ \pi$, so on the base $Xf=({\pi }_{\ast }X)f$. Hence $[X,fY]=f[X,Y]+(({\pi }_{\ast }X)f)Y$, the algebroid Leibniz rule 03.04.16. Therefore $(\mathrm{At}(P),[\cdot ,\cdot ],{\pi }_{\ast })$ is a Lie algebroid. $\square$

Bridge. This theorem identifies the middle term $\mathrm{At}(P)$ with the $G$-invariant vector fields on $P$ and the kernel $\mathrm{ad}(P)$ with the invariant vertical fields, so that the Atiyah sequence is the fibrewise statement that every infinitesimal move splits, after a choice, into a base push and a fibre twist; it builds toward the connection theorem below, where a splitting of this exact sequence is shown to be exactly a principal connection, and it appears again in the curvature formula, where the failure of a splitting to respect the bracket lands in $\Gamma (\mathrm{ad}(P))$. The foundational reason every structure map here is smooth is the freeness and properness of the principal action, which made $TP/G$ a vector bundle and $d\pi$ a constant-rank surjection. This is exactly the infinitesimal counterpart of the gauge groupoid 03.05.21: differentiating that groupoid along its units produces $\mathrm{At}(P)$, so the Atiyah algebroid generalises the Lie-algebra-of-a-Lie-group passage 03.04.01 to the bundle setting, and identifies the structure-group data with the kernel bundle of the anchor.

Exercises Intermediate+

Lean formalization Intermediate+

lean_status: none — Mathlib has a developing principal-bundle and smooth-manifold library (TangentSpace, ContMDiff sections, group actions) but no LieAlgebroid structure and no Atiyah-algebroid construction. 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.
variable {G M P : Type*} [LieGroup G] [Manifold M] [Manifold P]
variable (π : P → M) [PrincipalBundle G π]   -- free proper right action

-- lifted G-action on TP, quotient is the Atiyah algebroid
def AtiyahBundle := Quotient (liftedAction G (TangentBundle P))

-- sections ≅ G-invariant vector fields on P
def atiyahSections : Section (AtiyahBundle π) ≃ InvariantVectorFields G P := ...

def atiyahAlgebroid : LieAlgebroid M (AtiyahBundle π) where
  anchor  := fun X => pushforward π X            -- descends d π
  bracket := fun X Y => quotientBracket X Y      -- bracket of invariant fields
  -- kernel of anchor = ad(P) = (P × 𝔤)/G
  -- exact sequence 0 → ad(P) → AtiyahBundle → TM → 0
  ...

The first genuine obstacle is the quotient vector bundle: Mathlib lacks a packaged "quotient of $TP$ by a free proper lifted Lie-group action is a vector bundle over $P/G$" lemma, on which the smoothness of the anchor and the descended bracket both depend.

Advanced results Master

The Atiyah algebroid as the Lie algebroid of the gauge groupoid. Differentiating the gauge groupoid $(P\times P)/G\rightrightarrows M$ 03.05.21 along its unit submanifold produces a Lie algebroid, and that algebroid is $\mathrm{At}(P)=TP/G$ ^{[Mackenzie Ch. III §3]}. The source fibre of the gauge groupoid through a unit $1_x$ is diffeomorphic to $P$, and the tangent space at the unit to that source fibre, assembled over $M$, recovers $TP/G$; the target differential descends to the anchor ${\pi }_{\ast }$. This places the Atiyah sequence as the infinitesimal shadow of the gauge groupoid: the kernel $\mathrm{ad}(P)$ is the bundle of isotropy Lie algebras, differentiating the isotropy groups ${\Omega }_x^x\cong G$, and the bracket on $\Gamma (\mathrm{At}(P))$ is the differentiated form of source-fibre composition. The Atiyah algebroid is thus the bundle-theoretic instance of the Lie-functor passage from a Lie group to its Lie algebra 03.04.01, lifted to the transitive groupoid setting.

Connections as right-splittings; curvature as the bracket defect. A principal connection on $P$ 03.05.07 is the same datum as a vector-bundle splitting $\sigma :TM\to \mathrm{At}(P)$ with ${\pi }_{\ast }\circ \sigma ={\mathrm{id}}_{TM}$. Equivalently a connection is a left inverse $\omega :\mathrm{At}(P)\to \mathrm{ad}(P)$ to $\iota$, the connection one-form. The curvature is the obstruction to the splitting being a Lie-algebroid morphism, the $\mathrm{ad}(P)$-valued $2$-form $$ F(v, w) = \sigma[v, w] - [\sigma v, \sigma w] \in \Gamma(\mathrm{ad}(P)), $$ and a flat connection is exactly a splitting that is a bracket homomorphism, equivalently one whose horizontal distribution is involutive. Since the Atiyah sequence splits as vector bundles — one averages local splittings against a partition of unity, the average of vector-bundle splittings being a vector-bundle splitting — principal connections always exist. The existence question for flat connections is the harder one, governed by whether the bracket defect can be made to vanish.

The Chern-Weil shadow. The curvature $F\in {\Omega }^2(M;\mathrm{ad}(P))$ feeds the Chern-Weil homomorphism. An $\mathrm{Ad}$-invariant polynomial $p$ of degree $k$ on $\mathfrak{g}$ extends to a bundle map on $\mathrm{ad}(P)$, and $p(F)$ is a closed $2k$-form on $M$ whose de Rham class is independent of the connection. The independence is the algebroid statement that two splittings of the same Atiyah sequence differ by a section of $\mathrm{Hom}(TM,\mathrm{ad}(P))$, and the difference of their Chern-Weil forms is exact. The characteristic classes of $P$ are therefore invariants of the Atiyah sequence itself, read off any splitting.

Holomorphic obstruction (Atiyah's original setting). Atiyah introduced the sequence for a holomorphic principal bundle over a complex manifold, where the analogous short exact sequence of sheaves $$ 0 \to \mathrm{ad}(P) \to \mathrm{At}(P) \to TM \to 0 $$ is a sequence of holomorphic vector bundles, and a holomorphic connection is a holomorphic splitting ^{[Atiyah 1957]}. The obstruction to splitting is the extension class $a(P)\in H^1(M;\mathrm{ad}(P)\otimes {\Omega }_M^1)$, the Atiyah class, and for a line bundle it is the first Chern class in Hodge-theoretic guise. The smooth sequence always splits; the holomorphic one need not, and Atiyah's theorem is that a compact Kähler manifold admits a holomorphic connection on $P$ if and only if the Atiyah class vanishes.

Synthesis. The Atiyah algebroid $\mathrm{At}(P)=TP/G$ is the single vector bundle that records every $G$-invariant infinitesimal move of a principal bundle, and the central insight is that it converts the differential geometry of $P$ into the algebroid calculus of invariant vector fields: sections are the invariant fields, the anchor ${\pi }_{\ast }$ is the projection $d\pi$ descended, and the kernel $\mathrm{ad}(P)$ is the bundle of invariant vertical fields. The Atiyah sequence $0\to \mathrm{ad}(P)\to \mathrm{At}(P)\to TM\to 0$ is exact because the principal action is free and proper, and it is dual to the description of a connection: a right-splitting $\sigma$ is a principal connection, a left-splitting $\omega$ is its connection form, and putting these together gives the curvature $F=\sigma [\cdot ,\cdot ]-[\sigma \cdot ,\sigma \cdot ]$ as the failure of $\sigma$ to be a bracket morphism. This is exactly the infinitesimal counterpart of the gauge groupoid 03.05.21: the Atiyah algebroid is its Lie algebroid, $\mathrm{ad}(P)$ differentiates the isotropy, and the bracket differentiates source-fibre composition, so the bundle-theoretic and groupoid-theoretic pictures are one. The construction generalises the Lie-algebra-of-a-Lie-group passage 03.04.01 to the bundle setting and identifies the structure-group data with the kernel of the anchor; over the Chern-Weil homomorphism it identifies characteristic classes of $P$ with connection-independent invariants of the Atiyah sequence, and in Atiyah's holomorphic setting it identifies the obstruction to a holomorphic connection with the extension class of the sequence.

Full proof set Master

Proposition (the quotient $TP/G$ is a vector bundle and the structure maps descend). Let $\pi :P\to M$ be a principal $G$-bundle. The lifted $G$-action on $TP$ is free and proper; the quotient $TP/G$ is a smooth vector bundle over $M$ of rank $\dim M+\dim G$; the projection $TP\to TP/G$ is a surjective submersion; and the anchor ${\pi }_{\ast }$ and bracket descend to smooth structures.

Proof. The lifted action $h\cdot v=(d{R}_h)v$ is free: if $(d{R}_h)v=v$ with $v\in T_{p}P$, then $R_h$ fixes the base point $\pi (p\cdot h)=\pi (p)$ and, projecting to the fibre, $R_h$ fixes $p$, so $h=e$ by freeness of the principal action; here freeness on tangent vectors follows because $R_h$ is a diffeomorphism with no fixed points unless $h=e$. Properness is inherited from the properness of the $P$-action through the bundle projection $TP\to P$, which is $G$-equivariant: a convergent sequence $v_n\to v$, $(d{R}_{h_n})v_n\to w$ projects to $p_n\to p$, $p_n\cdot h_n\to {\pi }_{TP}(w)$, and properness of the $P$-action extracts a convergent subsequence of $(h_n)$. By the quotient theorem for free proper actions, $TP/G$ is a smooth manifold; the linear structure on the fibres of $TP$ is $G$-equivariant (each $d{R}_h$ is fibrewise linear), so $TP/G\to M$ inherits a vector-bundle structure with fibre $T_{p}P$ for a representative $p$, of dimension $\dim M+\dim G$. The anchor ${\pi }_{\ast }$ descends because $d\pi$ is $G$-invariant: $d\pi \circ d{R}_h=d(\pi \circ R_h)=d\pi$; it is fibrewise surjective because $\pi$ is a submersion, and of constant rank $\dim M$. The bracket descends because $\mathfrak{X}(P{)}^G$ is closed under the bracket, proved in the Key theorem. $\square$

Proposition (the kernel of the anchor is the adjoint bundle). Let $\pi :P\to M$ be a principal $G$-bundle. The fundamental-vector-field map descends to a vector-bundle isomorphism $\iota :\mathrm{ad}(P)=(P\times \mathfrak{g})/G\stackrel{\sim }{\to }\ker {\pi }_{\ast }$, where $G$ acts on $P\times \mathfrak{g}$ by $h\cdot (p,\xi )=(p\cdot h,{\mathrm{Ad}}_{h^{-1}}\xi )$.

Proof. Define $\Phi :P\times \mathfrak{g}\to TP$ by $\Phi (p,\xi )={\zeta }_{\xi }(p)=\frac{d}{dt}{|}_{0}p\cdot \exp (t\xi )$. For fixed $p$, $\xi \mapsto {\zeta }_{\xi }(p)$ is linear; it is injective because the action is free (a nonzero $\xi$ with ${\zeta }_{\xi }(p)=0$ would make $t\mapsto p\cdot \exp (t\xi )$ stationary, contradicting freeness on the one-parameter subgroup), and its image is the vertical space $V_p=\ker (d{\pi }_p)$, of dimension $\dim G=\dim \mathfrak{g}$, so $\xi \mapsto {\zeta }_{\xi }(p)$ is an isomorphism $\mathfrak{g}\stackrel{\sim }{\to }{V}_p$. Equivariance: differentiating the identity $R_h(p\cdot \exp (t\xi ))=(p\cdot h)\cdot \exp \left(t{\mathrm{Ad}}_{h^{-1}}\xi \right)$ at $t=0$ gives $(d{R}_h){\zeta }_{\xi }(p)={\zeta }_{{\mathrm{Ad}}_{h^{-1}}\xi }(p\cdot h)$, which is exactly the statement that $\Phi$ intertwines $h\cdot (p,\xi )=(p\cdot h,{\mathrm{Ad}}_{h^{-1}}\xi )$ with the lifted action on $TP$. Quotienting by $G$, $\Phi$ descends to a fibrewise-isomorphic bundle map $\iota :(P\times \mathfrak{g})/G\to V/G\subseteq TP/G$, and $V/G=\ker {\pi }_{\ast }$ because $V=\ker d\pi$. Hence $\iota$ is an isomorphism onto $\ker {\pi }_{\ast }$. $\square$

Proposition (connections are exactly right-splittings, bijectively). Let $\pi :P\to M$ be a principal $G$-bundle with Atiyah sequence $0\to \mathrm{ad}(P)\to \mathrm{At}(P)\stackrel{{\pi }_{\ast }}{\to }TM\to 0$. The assignment $H\mapsto {\sigma }_H$ sending a principal connection (a $G$-invariant horizontal complement $H$ to the vertical bundle) to its horizontal-lift splitting is a bijection onto the set of vector-bundle splittings $\sigma :TM\to \mathrm{At}(P)$ of ${\pi }_{\ast }$.

Proof. Given a principal connection $H\subseteq TP$, the map $d\pi :H_p\to T_{\pi (p)}M$ is a linear isomorphism, and the horizontal lift ${\tilde{v}}_p\in H_p$ of $v$ satisfies ${\tilde{v}}_{p\cdot h}=(d{R}_h){\tilde{v}}_p$ by invariance of $H$; so $p\mapsto {\tilde{v}}_p$ is a $G$-invariant vector field and defines ${\sigma }_H(v)\in \mathrm{At}(P)$ with ${\pi }_{\ast }{\sigma }_H(v)=v$. The map ${\sigma }_H$ is linear in $v$ since $d\pi {|}_H$ is a linear isomorphism. Conversely, given a splitting $\sigma$, lift each $\sigma (v)\in \mathrm{At}(P)$ to its invariant vector field on $P$; the subspace $H_p:=\{\sigma (v{)}_p:v\in T_{\pi (p)}M\}$ is a $G$-invariant complement to $V_p$ because $d\pi \circ \sigma =\mathrm{id}$ forces $H_p\cap V_p=0$ and a dimension count gives $H_p\oplus V_p=T_{p}P$. These two assignments are mutually inverse: ${\sigma }_{H_{\sigma }}=\sigma$ because the horizontal lift of $v$ in $H_{\sigma }$ is $\sigma (v)$, and $H_{{\sigma }_H}=H$ because the image of ${\sigma }_H$ recovers the horizontal subspaces. Hence the correspondence is a bijection. $\square$

Connections Master

The gauge groupoid 03.05.21 is the global object the Atiyah algebroid integrates: differentiating $(P\times P)/G$ along its units returns $TP/G$, with the kernel $\mathrm{ad}(P)$ differentiating the isotropy groups ${\Omega }_x^x\cong G$ and the bracket differentiating source-fibre composition. The Atiyah sequence is therefore the infinitesimal form of Mackenzie's correspondence between transitive Lie groupoids and principal bundles, read one derivative down.

The Lie algebroid 03.04.16 supplies the ambient framework: $\mathrm{At}(P)$ is the flagship transitive example, where the anchor ${\pi }_{\ast }$ is surjective and its kernel is the bundle of isotropy Lie algebras. The general theorem that the anchor of a Lie algebroid is automatically a bracket homomorphism specialises here to the statement that ${\pi }_{\ast }$ intertwines the bracket of invariant fields on $P$ with the vector-field bracket on $M$, which is what makes curvature land in $\mathrm{ad}(P)$.

A principal bundle 03.05.01 is the input to the construction, and its associated bundles surface as the kernel: the adjoint bundle $\mathrm{ad}(P)=(P\times \mathfrak{g})/G$ is the bundle associated to the adjoint representation, so the algebroid's kernel is determined by the structure-group data alone, independent of any connection.

A principal connection 03.05.07 is, at the algebroid level, a right-splitting of the Atiyah sequence; curvature is the failure of that splitting to preserve the bracket, so the curvature theory of connections is the obstruction theory for lifting the anchor to a Lie-algebroid morphism, and the flat case is the involutive (integrable) splitting.

Historical & philosophical context Master

The infinitesimal exact sequence now bearing Atiyah's name appeared in Michael Atiyah's 1957 study of complex analytic connections in fibre bundles, where, for a holomorphic principal bundle over a complex manifold, the obstruction to a holomorphic connection is the failure of a short exact sequence of sheaves to split holomorphically, measured by an extension class — the Atiyah class — living in a first sheaf-cohomology group ^{[Atiyah 1957]}. Atiyah's motivation was the existence of holomorphic connections, and his sequence is the holomorphic precursor of the smooth Atiyah algebroid. The smooth, differential-geometric reading of the same sequence, with horizontal distributions and curvature, traces to Ehresmann's 1950 formulation of infinitesimal connections on differentiable fibre bundles ^{[Ehresmann 1950]}.

The algebroid synthesis — $TP/G$ as a Lie algebroid, the sequence as its anchor sequence, connections as splittings, and the whole structure as the Lie algebroid of the gauge groupoid — is due to Kirill Mackenzie, whose 1987 lecture notes and 2005 monograph made the Atiyah sequence the organising device for connection theory in the groupoid-algebroid framework ^{[Mackenzie Ch. III §3]}. The construction also appears in Bertram Kostant's 1970 work on quantization, where the Atiyah sequence of a circle bundle encodes the prequantum data and the connection is the prequantum connection ^{[Kostant 1970]}. The sequence sits at the meeting point of Atiyah's analytic program and Ehresmann's bundle-theoretic one, and its name records the holomorphic origin even though its most common use is now the smooth theory of gauge fields.

Bibliography Master

@article{atiyah1957,
  author    = {Atiyah, Michael F.},
  title     = {Complex analytic connections in fibre bundles},
  journal   = {Transactions of the American Mathematical Society},
  volume    = {85},
  pages     = {181--207},
  year      = {1957}
}

@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}
}

@incollection{kostant1970,
  author    = {Kostant, Bertram},
  title     = {Quantization and unitary representations},
  booktitle = {Lectures in Modern Analysis and Applications III},
  series    = {Lecture Notes in Mathematics},
  volume    = {170},
  pages     = {87--208},
  publisher = {Springer},
  year      = {1970}
}

@incollection{ehresmann1950,
  author    = {Ehresmann, Charles},
  title     = {Les connexions infinit\'esimales dans un espace fibr\'e diff\'erentiable},
  booktitle = {Colloque de Topologie (Espaces Fibr\'es), Bruxelles, 1950},
  pages     = {29--55},
  publisher = {Georges Thone, Li\`ege; Masson, Paris},
  year      = {1951}
}

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Atiyah algebroid — the Lie algebroid $\mathrm{At}(P)=TP/G\to M$ of a principal bundle, with sections the $G$-invariant vector fields, anchor ${\pi }_{\ast }$, kernel the adjoint bundle, and the Atiyah sequence $0\to \mathrm{ad}(P)\to TP/G\to TM\to 0$ whose splittings are connections and whose splitting defect is curvature; the infinitesimal counterpart of the gauge groupoid.