Atiyah sequence

Let $\pi:P\to M$ be a principal $G$-bundle with Lie algebra $\mathfrak{g}$. Its Atiyah algebroid is $A(P)=TP/G\to M$, equipped with the anchor $a:A(P)\to TM$ induced by $d\pi$.

Define the adjoint bundle $\mathrm{ad}(P)=P\times_{\mathrm{Ad}}\mathfrak{g}\to M$. There is a natural injection

defined fiberwise by sending $[p,X]\in \mathrm{ad}(P)_x$ to the class of the fundamental vertical vector $X^\#_p\in T_pP$. The anchor $a$ is surjective, and its kernel is exactly the image of $\iota$. Thus one obtains the Atiyah sequence of vector bundles over $M$:

Exactness means:

• $\iota$ is injective,
• $a\circ \iota=0$,
• $\ker(a)=\mathrm{im}(\iota)$,
• $a$ is surjective.

Examples

1. Trivial bundle. If $P=M\times G$, then $TP/G\cong TM\oplus (M\times\mathfrak{g})$, $\mathrm{ad}(P)\cong M\times \mathfrak{g}$, and the sequence is

   $$0\to M\times\mathfrak{g}\to TM\oplus(M\times\mathfrak{g})\to TM\to 0,$$

   split by the obvious projection.

2. Bundle over a point. For $P=G\to\{\ast\}$, the sequence becomes $0\to \mathfrak{g}\xrightarrow{=}\mathfrak{g}\to 0$.

3. Circle bundles. For a principal $U(1)$-bundle, $\mathrm{ad}(P)\cong M\times i\mathbb{R}$ is a trivial line bundle, and the sequence exhibits $TP/U(1)$ as an extension of $TM$ by this line.