Atiyah algebroid
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"Atiyah sequence" redirects here. For the spectral sequence of Atiyah and Hirzebruch, see Atiyah–Hirzebruch spectral sequence.
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.[1] It plays a crucial example in the integrability of (transitive) Lie algebroids, and it has applications in gauge theory and geometric mechanics.