Atiyah sequence - Knowlpedia

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

\iota:\mathrm{ad}(P)\hookrightarrow TP/G

defined fiberwise by sending $[p,X]\in \mathrm{ad}(P)_x$ to the class of the fundamental vertical vector $X^\#_p\in T_pP$ . This sign agrees with the Lie-algebroid convention in the linked Atiyah algebroid: invariant vertical vector fields along a group fiber are right-invariant, whose bracket is the negative of the usual Lie-algebra bracket. The anchor $a$ is surjective, and its kernel is exactly the image of $\iota$ . Thus one obtains the Atiyah sequence of vector bundles over $M$ :

0 \longrightarrow \mathrm{ad}(P) \xrightarrow{\ \iota\ } TP/G \xrightarrow{\ a\ } \, TM \longrightarrow 0.

Exactness means:

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

Examples

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 inclusion $v\mapsto(v,0)$ .

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

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.