Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 1 / 3 / c

A smooth vector field on a smooth manifold $M$ is a smooth section of its tangent bundle: it assigns $v(p)\in T_{p}M$ smoothly to each point. On a Lie group, it is a left-invariant vector field whenFor $X\in T_{e}G$, define $v_X(g)=(d{L}_g{)}_{e}X$. Smoothness of multiplication makes this a smooth vector field, and $L_g{L}_p=L_{gp}$ plus the chain rule proves left invariance. Every left-invariant vector field arises this way from its value at the identity.

For a matrix Lie group, $v_X(g)=gX$, because the derivative at zero of $g\gamma (t)$ is $gX$. The printed group name in this subpart must be read as ${\mathrm{SL}}_2$, rather than the additive vector space ${\mathfrak{sl}}_2$: it is the matrix-group calculation consistent with the preceding subparts. At the specified matrix,A tangent vector at a nonidentity group element need not itself be trace-free; here $g^{-1}{v}_X(g)=X$ is trace-free, as required. If one instead interpreted the printed ${\mathfrak{sl}}_2$ as its additive Lie group, the field would be the constant field $X$, a different problem.