Recall that an object in an abelian category is semisimple if it is a (finite) direct sum of simple objects. An abelian category is semisimple if every object is semisimple.
In studying representations of semisimple lie algebras, one often restricts to the subcategory called $\mathcal{O}_{int}$ inside the category left modules for the universal enveloping algebra, $\mathcal{U}(\mathfrak{g})$. The main theorem about $\mathcal{O}_{int}$ says it is semisimple (or, at least, that all the objects in $\mathcal{O}_{int}$ are semisimple. Perhaps one needs to check several other things to see that $\mathcal{O}_{int}$ is abelian as a subcategory).
I am new to this area and would like to motivate this restriction.
A ring $R$ is semisimple in the category of left $R$ modules iff the category of left $R$ modules is semisimple. Thus,
Is $\mathcal{U}(\mathfrak{g})$ semisimple as a left $\mathcal{U}(\mathfrak{g})$-module (for $\mathfrak{g}$ a semisimple Lie algebra)?
As a bonus, are there other nice examples of $\mathcal{U}(\mathfrak{g})$-modules that aren't semisimple?
Best Answer
If $U(\mathfrak{g})$ were semi-simple as a $U(\mathfrak{g})$-module, then the same would be true of any quotient. However, $U(\mathfrak{g})/U(\mathfrak{g})\mathfrak{b}\cong M(0)$, where $M(0)$ is the Verma module of highest weight $0$, $\mathfrak{b}=\mathfrak{h}\oplus\mathfrak{n}^+$ and $\mathfrak{g}=\mathfrak{n}^-\oplus\mathfrak{h}\oplus\mathfrak{n}^+$ is a triangular decomposition of $\mathfrak{g}$.
It is well known, and easy to check that $M(0)$ is not semi-simple. Try it for $\mathfrak{sl}_2$.