TITLE: What is the categorical significance of the trivial $\mathfrak{g}$-module in the category of $\mathfrak{g}$-mod?
QUESTION [8 upvotes]: This question may be trivial for experts.
Let $\mathfrak{g}$ be a Lie algebra over a field $k$ and consider the category $\mathfrak{g}$-mod of $\mathfrak{g}$ modules. We can add suitable conditions, like finite dimension, semisimple, if necessary.
We call the 1-dimensional $k$-space with  $\mathfrak{g}$ acts by $0$ the trivial $\mathfrak{g}$-module. This $\mathfrak{g}$ module is kind of special in the category $\mathfrak{g}$-mod. But it is not the initial or terminal object (the $0$-module is both initial and terminal).
$\textbf{My question}$ is: is there any categorical significance of this trivial $\mathfrak{g}$ module? Do we need to take the tensor structure into account?

REPLY [4 votes]: To address the category O issue: no.  Every finite dimensional simple in category O is "the same," in the sense there's a functor sending one to the other which isn't an equivalence on all of category O (you could cook such a thing up, but it wouldn't be very natural), but is an equivalence between the blocks of the two corresponding simples: the translation functors do this.