Dual Lie algebra representation
= Dual Lie algebra representation
{title2=$\rho^*(x)=-\rho(x)^T$}
The dual of a <Lie algebra representation> on $V$ acts on its <dual space> by $(x\phi)(v)=-\phi(xv)$. The minus sign makes this a <Lie algebra homomorphism>: taking transposes reverses the order in a <commutator>. It is the infinitesimal version of the <dual representation> of a <group>.