Let $\mathfrak g$ be a Lie algebra and let $(V,\rho_V)$ be a representation of $\mathfrak g$. Then the dual representation of $\mathfrak g$ is the pair $(V^,\rho_{V^})$ where $V^$ is the dual space of $V$ and $\rho_{V^}$ is given by \(\rho_{V^*}(x) = -\rho_V(x)^ *\) for all $x \in \mathfrak g$.
Wikidata ID: Q3427408