Dual Lie algebra representation (${\rho }^{\ast }(x)=-\rho (x{)}^T$)

The dual of a Lie algebra representation on $V$ acts on its dual space by $(x\varphi )(v)=-\varphi (xv)$. 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.

For a finite-dimensional irreducible representation $L(\lambda )$ of a complex semisimple Lie algebra, its dual Lie algebra representation has highest weight $-w_0\lambda$, where $w_0$ is the longest Weyl-group element. Indeed duality negates all weights, and $w_0\lambda$ is the lowest weight of the original module. In particular $w_0=-I$ makes every such irreducible module self-dual.