= Pauli-coordinate left-invariant vector fields on SU(2)
{c}
{title2=$T_j=i(u_0\delta_{ij}+u_k\epsilon_{jki})\partial_{u_i}$}
On a hemisphere of <SU(2) as the three-sphere>, write $A=u_0I+i u_i\sigma_i$, with real coefficients satisfying $u_0^2+|\mathbf u|^2=1$. Right multiplication by $I+i v_j\sigma_j$ gives $du_i=(u_0\delta_{ij}+u_k\epsilon_{jki})v_j$. This right multiplication flow defines left-invariant tangent fields. With the displayed factor $i$, they obey $T_jA=-A\sigma_j$, since $\partial_{u_i}u_0=-u_i/u_0$. Applying the <commutator> to $A$ and using the <Pauli matrix commutator identity> gives $[T_i,T_j]=-2i\epsilon_{ijk}T_k$. The entries of $A$ contain every coordinate, so this determines the vector-field identity, not just its action in one isolated representation.
Back to article page
