AtiyahAlgebroiden

The Atiyah algebroiden, or Atiyah algebroid, is a construction in differential geometry associated to a principal G-bundle P → M. It is the Lie algebroid that encodes infinitesimal symmetries of the bundle and can be realized concretely as the quotient TP/G, where TP is the tangent bundle of P equipped with the natural G-action.

As a vector bundle over M, the Atiyah algebroid A(P) comes with a Lie bracket on its space of sections and an anchor map to the tangent bundle TM. The anchor is induced by the differential of the bundle projection π: P → M, sending a class of tangent vectors on P to the corresponding tangent vector on M. There is a canonical short exact sequence of vector bundles 0 → ad(P) → A(P) → TM → 0, where ad(P) ≅ P ×_G g is the adjoint bundle formed from the Lie algebra g of G by the adjoint action. The kernel ad(P) consists of the vertical G-invariant vector fields along the fibers, i.e., infinitesimal gauge transformations.

The Lie algebroid structure on A(P) is inherited from the Lie bracket of G-invariant vector fields on P: the bracket on sections descends to A(P), and the anchor intertwines this bracket with the usual Lie bracket of vector fields on M. A connection on P corresponds to a splitting of the exact sequence, yielding a decomposition A(P) ≅ TM ⊕ ad(P). The curvature of the connection measures the obstruction to integrability of the splitting.

For vector bundles E, the Atiyah algebroid of E is obtained by taking P to be the frame bundle Fr(E) with structure group GL(n). In this case ad(P) corresponds to End(E), and the sequence becomes 0 → End(E) → A(E) → TM → 0. The Atiyah algebroid thus generalizes the notion of infinitesimal symmetries and connections in gauge theory and serves as a natural Lie algebroid framework for studying differential operators and characteristic classes on bundles.