Atiyah algebroid

In mathematics, the Atiyah algebroid, or Atiyah sequence, of a principal ${\displaystyle G}$-bundle ${\displaystyle P}$ over a manifold ${\displaystyle M}$, where ${\displaystyle G}$ is a Lie group, is the Lie algebroid of the gauge groupoid of ${\displaystyle P}$. Explicitly, it is given by the following short exact sequence of vector bundles over ${\displaystyle M}$:

${\displaystyle 0\to P\times _{G}{\mathfrak {g}}\to TP/G\to TM\to 0.}$

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.