assume that $\mathfrak{g}$ is a finite dimensional Lie algebra. Is rather well-known that its corresponding tensor algebra $T(\mathfrak{g})$ admits a $\mathfrak{g}$-module structure. I have been looking for this $\mathfrak{g}$-module structure, but seems to be different from what I was expecting. For instance I thought that the "natural" action should work (i.e $g . (u_1 \otimes u_2 \otimes...\otimes u_n)= (g.u_1) \otimes (g. u_2 )\otimes ... \otimes (g. u_n))$. But turns out I think after spending sometime staring it, that isn't quite correct. My books unfortunately doesn't contain any comment about this action (perhaps must be the only one to whom isn't obvious in a first glance :)). Though, I think that a fair action could be something like $g . (u_1 \otimes u_2 \otimes...\otimes u_n) = \sum u_1 \otimes u_2 \otimes ... \otimes (g.u_j) \otimes ... \otimes u_n$, where the leftover terms probably cancel each other out. Can you please give me a reference, or even better explain me if the aforementioned is correct or not? Also is it true that the "natural" action doesn't work in that case?
I forgot to mention that the $\mathfrak{g}$-module structure on $\mathfrak{g}$, is given by the usual adjoint action, i.e. $g.u_j=[g,u_j]$.
Thank you!
You probably want the action to be linear, i.e $(\lambda g) \cdot x = \lambda \cdot (g \cdot x)$ for all $\lambda \in k$. With your definition, $(\lambda g) \cdot x = \lambda^n g \cdot x$.
The action of $\mathfrak g$ on $T(\mathfrak g)$ is usually defined on $2$-tensor by $g. (u \otimes v) = g.u \otimes v + u \otimes gv$ which is equivalent to what you wrote.