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.
defines a short exact sequence: of vector bundles over
acts on the vector bundles in this sequence.
is isomorphic to the trivial vector bundle
action is the adjoint bundle
of the exact sequence above yields a short exact sequence:
, which is called the Atiyah sequence of
has an associated Lie groupoid, called its gauge groupoid, whose objects are points of
, and whose morphisms are elements of the quotient of
, with source and target given by the two projections of
By definition, the Atiyah algebroid of
Last, the kernel of the anchor is isomorphic precisely to
-modules by taking the space of sections of the vector bundles.
More precisely, the sections of the Atiyah algebroid of
under Lie bracket, which is an extension of the Lie algebra of vector fields on
-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.
The Atiyah algebroid of a principal
is always: Note that these two properties are independent.
Integrable Lie algebroids does not need to be transitive; conversely, transitive Lie algebroids (often called abstract Atiyah sequences) are not necessarily integrable.
While any transitive Lie groupoid is isomorphic to some gauge groupoid, not all transitive Lie algebroids are Atiyah algebroids of some principal bundle.
Integrability is the crucial property to distinguish the two concepts: a transitive Lie algebroid is integrable if and only if it is isomorphic to the Atiyah algebroid of some principal bundle.
of the Atiyah sequence of a principal bundle
are in bijective correspondence with principal connections on
Similarly, the curvatures of such connections correspond to the two forms
of principal bundles induces a Lie algebroid morphism
between the respective Atiyah algebroids.