Atiyah algebroid

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

It is named after Michael Atiyah, who introduced the construction to study the existence theory of complex analytic connections. It plays a crucial example in the integrability of (transitive) Lie algebroids, and it has applications in gauge theory and geometric mechanics.

Definitions

As a sequence

For any fiber bundle ${\displaystyle P}$ over a manifold ${\displaystyle M}$, the differential ${\displaystyle d\pi }$ of the projection ${\displaystyle \pi :P\to M}$ defines a short exact sequence:

${\displaystyle 0\to VP\to TP\stackrel{d\pi }{\to }{\pi }^{\ast }TM\to 0}$

of vector bundles over ${\displaystyle P}$, where the vertical bundle ${\displaystyle VP}$ is the kernel of ${\displaystyle d\pi }$.

If ${\displaystyle P}$ is a principal ${\displaystyle G}$-bundle, then the group ${\displaystyle G}$ acts on the vector bundles in this sequence. Moreover, since the vertical bundle ${\displaystyle VP}$ is isomorphic to the trivial vector bundle ${\displaystyle P\times \mathfrak{g}\to P}$, where ${\displaystyle \mathfrak{g}}$ is the Lie algebra of ${\displaystyle G}$, its quotient by the diagonal ${\displaystyle G}$ action is the adjoint bundle ${\displaystyle P{\times }_G\mathfrak{g}}$. In conclusion, the quotient by ${\displaystyle G}$ of the exact sequence above yields a short exact sequence:$${\displaystyle 0\to P{\times }_G\mathfrak{g}\to TP/G\to TM\to 0}$$ of vector bundles over ${\displaystyle P/G\cong M}$, which is called the Atiyah sequence of ${\displaystyle P}$.

As a Lie algebroid

Recall that any principal ${\displaystyle G}$-bundle ${\displaystyle P\to M}$ has an associated Lie groupoid, called its gauge groupoid, whose objects are points of ${\displaystyle M}$, and whose morphisms are elements of the quotient of ${\displaystyle P\times P}$ by the diagonal action of ${\displaystyle G}$, with source and target given by the two projections of ${\displaystyle M}$. By definition, the Atiyah algebroid of ${\displaystyle P}$ is the Lie algebroid ${\displaystyle A\to M}$ of its gauge groupoid.

It follows that ${\displaystyle A=TP/G}$, while the anchor map ${\displaystyle A\to TM}$ is given by the differential ${\displaystyle d\pi :TP\to TM}$, which is ${\displaystyle G}$-invariant. Last, the kernel of the anchor is isomorphic precisely to ${\displaystyle P{\times }_G\mathfrak{g}}$.

The Atiyah sequence yields a short exact sequence of ${\displaystyle {\mathcal{C}}^{\mathrm{\infty }}(M)}$-modules by taking the space of sections of the vector bundles. More precisely, the sections of the Atiyah algebroid of ${\displaystyle P}$ is the Lie algebra of ${\displaystyle G}$-invariant vector fields on ${\displaystyle P}$ under Lie bracket, which is an extension of the Lie algebra of vector fields on ${\displaystyle M}$ by the ${\displaystyle G}$-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.