Proposition 3

Let $g$ be a Lie algebra over $K$ and $M_i$ a $g$-module ($1 \leq i \leq n + 1$). Let $N$ be the $K$-module $\mathscr{L}_K(M_1, \ldots, M_n; M_{n+1})$ of multilinear mappings of $\prod_{i=1}^n M_i$ into $M_{n+1}$. There exists one and only one $g$-module structure on $N$ such that

for all $x \in g, u \in N$ and $m_i \in M_i$ ($1 \leq i \leq n$).

In particular, let $g$ be a Lie algebra over $K$ and $M$ a $g$-module and consider $K$ as a trivial $g$-module. Proposition 3 defines a $g$-module structure on $\mathscr{L}_K(M, K) = M^*$. The corresponding representation is called the dual representation of the representation $x \mapsto x_M$. We have:

for all $x \in g, f \in M^*, m \in M$. In other words:

When $K$ is a field and $M$ is finite-dimensional, the $g$-module $M$ is simple (resp. semi-simple) if and only if the $g$-module $M^*$ is simple (resp. semi-simple).