Atiyah algebroid

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The Atiyah algebroid (also called the Atiyah sequence) is a simple geometric object attached to a principal G-bundle P → M. It encodes both the gauge symmetries and the possible connections on P.

Construction and basic facts
- Start with a principal G-bundle P → M, where G is a Lie group. Look at the tangent bundle TP and the projection π: P → M.
- There is a short exact sequence of vector bundles over P: 0 → VP → TP → π* TM → 0, where VP is the vertical bundle (the kernel of dπ).
- For a principal G-bundle, G acts on these bundles. The vertical bundle VP is isomorphic to the trivial bundle P × g → P, where g is the Lie algebra of G. When you mod out by the diagonal G-action, you get the adjoint bundle P ×_G g.
- The quotient TP/G is a vector bundle over P/G ≅ M, and you obtain a short exact sequence of vector bundles over M:
0 → P ×_G g → TP/G → TM → 0.
This exact sequence is the Atiyah sequence of P.

Gauge groupoid viewpoint
- Every principal G-bundle P → M has an associated gauge groupoid. Its objects are points of M, and its morphisms are roughly P × P modulo the diagonal G-action.
- The Atiyah algebroid A → M is the Lie algebroid of this gauge groupoid. Concretely, A ≅ TP/G, and its anchor map A → TM is given by dπ: TP → TM (which is G-invariant).
- The kernel of the anchor map is precisely P ×_G g.

Sections and algebra
- The sections of the Atiyah algebroid A are the G-invariant vector fields on P, with the usual Lie bracket. They form an extension of the Lie algebra of vector fields on M by the G-invariant vertical fields.
- This can be viewed as an exact sequence of C∞(M)-modules (or of sheaves of local sections).

Integrability and transitivity
- Not every transitive Lie algebroid is integrable, and not every transitive Lie algebroid comes from a principal bundle in the Atiyah sense.
- Integrability means the algebroid comes from (is isomorphic to) the Atiyah algebroid of some principal bundle; this is a key way to distinguish Atiyah algebroids from general transitive algebroids.

Connections and curvature
- A right splitting σ: TM → A of the Atiyah sequence (TM → A → TM) corresponds to a principal connection on P → M.
- The curvature of this connection is the 2-form Ωσ ∈ Ω^2(M, P[g]) defined by
Ωσ(X, Y) = [σ(X), σ(Y)] − σ([X, Y]),
where [ , ] denotes the Lie bracket in A.

Morphisms
- If φ: P → P' is a morphism of principal bundles, it induces a Lie algebroid morphism dφ: TP/G → TP'/G' between the corresponding Atiyah algebroids.

In short, the Atiyah algebroid A → M captures the geometry of a principal G-bundle by combining its tangent directions on M with the vertical, gauge-related directions coming from g, organized as a short exact sequence that reflects connections and curvature.