Atiyah algebroid TP/G and its anchor

Let $\pi:P\to M$ be a principal G-bundle with structure group $G$. The right action of $G$ on $P$ differentiates to a right action on the tangent bundle $TP$ by $dR_g:TP\to TP$.

Construction (Atiyah algebroid). Form the quotient

the bundle whose fiber over $x\in M$ is the set of $G$-orbits in $T_pP$ for any $p\in P_x$. This is a smooth vector bundle over $M$.

Anchor map. The differential $d\pi:TP\to TM$ is $G$-invariant (since $\pi\circ R_g=\pi$), hence it descends to a vector bundle map

called the anchor.

Lie algebroid structure. Sections of $\mathrm{At}(P)$ identify with $G$-invariant vector fields on $P$. The usual Lie bracket of $G$-invariant vector fields is again $G$-invariant, so it defines a bracket on sections of $\mathrm{At}(P)$, making $\mathrm{At}(P)$ a Lie algebroid with anchor $a$.

There is a natural short exact sequence of vector bundles

where $\mathrm{ad}(P)$ embeds as the quotient of vertical tangent vectors (fundamental fields). A principal connection is equivalently a splitting of this sequence.

Examples

1. If $P$ is trivial, $P\cong M\times G$, then $\mathrm{At}(P)\cong TM\oplus (M\times \mathfrak g)$, and the anchor is projection onto $TM$.
2. For an abelian group, the bracket on the $\mathfrak g$-summand is zero, and $\mathrm{At}(P)$ is (locally) a semidirect product Lie algebroid determined only by how vertical and horizontal parts interact.
3. A choice of principal connection identifies $\mathrm{At}(P)$ with $TM\oplus \mathrm{ad}(P)$ by sending a class $[v_p]$ to its base component and its vertical component measured by the connection form.