Induced action from $SL$ to $\mathfrak sl$ Suppose $SL(V, \mathbb C)$ acts on $V$ as expected, and thus on $V \otimes (V\wedge V)$ as $g \cdot [ v \otimes (w \wedge z)] = g \cdot v \otimes (g \cdot w \wedge g \cdot z )$. Which is the induced action of $\mathfrak sl_n$ on this same spaces?

The action of $\mathfrak{sl}(V)$ on $V^{\otimes n}$ is given by $$g\cdot (v_1\otimes \dots\otimes v_n)=gv_1\otimes v_2\otimes\dots \otimes v_n+\dots+ v_1 \otimes \dots  \otimes v_{n-1}\otimes gv_n;$$
and hence the action on its quotients modules (such as $V^{\otimes (n-k)}\otimes\Lambda^k V$) has the same form.
It's obvious from differentiation of the group action (to get at least in a non-rigorous "physicist" way the formula, start from the group action $x\cdot v$, and then compute $(1+g)\cdot v-v$ where $g$ is thought to be infinitesimal and remove terms where $g$ appears at least twice). 
