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Concept
Atiyah algebroid
Résumé
In mathematics, the Atiyah algebroid, or Atiyah sequence, of a principal -bundle over a manifold , where is a Lie group, is the Lie algebroid of the gauge groupoid of . Explicitly, it is given by the following short exact sequence of vector bundles over :
It is named after Michael Atiyah, who introduced the construction to study the existence theory of complex analytic connections. It plays a crucial example in the integrability of (transitive) Lie algebroids, and it has applications in gauge theory and geometric mechanics.
For any fiber bundle over a manifold , the differential of the projection defines a short exact sequence
of vector bundles over , where the vertical bundle is the kernel of .
If is a principal -bundle, then the group acts on the vector bundles in this sequence. Moreover, since the vertical bundle is isomorphic to the trivial vector bundle , where is the Lie algebra of , its quotient by the diagonal action is the adjoint bundle . In conclusion, the quotient by of the exact sequence above yields a short exact sequence of vector bundles over , which is called the Atiyah sequence of .
Recall that any principal -bundle has an associated Lie groupoid, called its gauge groupoid, whose objects are points of , and whose morphisms are elements of the quotient of by the diagonal action of , with source and target given by the two projections of . By definition, the Atiyah algebroid of is the Lie algebroid of its gauge groupoid.
It follows that , while the anchor map is given by the differential , which is -invariant. Last, the kernel of the anchor is isomorphic precisely to .
The Atiyah sequence yields a short exact sequence of -modules by taking the space of sections of the vector bundles. More precisely, the sections of the Atiyah algebroid of is the Lie algebra of -invariant vector fields on under Lie bracket, which is an extension of the Lie algebra of vector fields on by the -invariant vertical vector fields. In algebraic or analytic contexts, it is often convenient to view the Atiyah sequence as an exact sequence of sheaves of local sections of vector bundles.
Source officielle
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