Indecomposable modules in the representation theory of Lie algebras

Question

I am relatively new to the representation theory of Lie algebras (my background to date has been primarily in the representation theory of finite dimensional associative algebras), and I was wondering about the following.

When one looks at the representation theory of associative algebras, the simple modules are often easy to describe (especially in the finite dimensional setting). They are also (somewhat) boring to consider in the sense that they don't provide a lot of information about the module category of the algebra. Thus, one often looks at the next biggest 'building block' of the module category, which is the indecomposable modules. This brings up all sorts of questions, such as the representation type of the algebra, and if it is tame (as opposed to wild), what form the indecomposables take. From there, one naturally considers the Hom spaces and one may attempt to give a description of these. And of course, we also have the Ext groups to consider and a description of these, which moves towards the homological side of things with attempting to describe the exact sequences. (And so on...)

Moving over to the non-associative side of things (specifically towards Lie algebras), the process of understanding the representation theory (as I have gathered from my reading so far) appears to be very different. It seems to me that there is a much greater focus on the irreducible representations of the algebra (which I guess are analogous to simple modules). In this setting, they appear to be much more interesting and their role in the representation theory of the algebra is more important. Much of the work I have seen thus far for various (not necessarily finite dimensional) Lie algebras has this focus on the irreducible representations and very little on the indecomposable representations in general.

For semisimple Lie algebras, I can understand this. The category of finite dimensional representations is semisimple, meaning that the indecomposable representations are precisely the irreducible ones (this is Weyl's Theorem). Looking at finite dimensional Lie algebras in general (so not necessarily semisimple), I guess I can also understand this, since most of them are wild (c.f. [1]). I guess also the Hom-spaces in the semisimple setting aren't very interesting due to Shur's Lemma, and hopeless to describe in generality in the wild setting?

What about in the setting of infinite dimensional Lie algebras? For particular infinite dimensional Lie algebras (for example twisted and untwisted loop algebras, current algebras, equivariant map algebras etc), I again often see this focus on irreducible representations and very little on the indecomposables in general (even when the category of f.d. representations for a Lie algebra is known to not be semisimple). I see very little work on discussing the representation type and very little work on determining the Hom-spaces. I've seen some work on Ext groups for some Lie algebras, but it hasn't been a particularly broad theme from my reading so far. Why is all of this the case? Is it a mainly historical tendency to not consider these things and focus more on irreducibles instead, or is there a particular mathematical reason for this? Is it just too difficult? (Or perhaps the answer is I need to do more reading, in which case I would appreciate references for the examples of infinite dimensional algebras I have given!)

[1] https://arxiv.org/abs/1202.1401

Answer 1

It might help you to shift your focus from the Lie algebra $\mathfrak{g}$ to its enveloping algebra $U=U(\mathfrak{g})$, which is an (infinite dimensional) associative algebra whose module category is equivalent to the category of representations of $\mathfrak{g}$.

Even classifying the simple $U$-modules is not simple (even for $\mathfrak{sl}_2$!). It's very closely analogous to the problem of classifying the simple modules over a Weyl algebra (and in fact, one of the main tools for studying them, Beilinson-Bernstein localization translates the problem into a problem about $D$-modules on $\mathbf{P}^1(\mathbf{C})$, in the case of $\mathfrak{sl}_2$). So, historically, people have focused on classifying and studying the simple objects in certain subcategories, such as the category $\mathcal{O}$ mentioned in the comments.

In that case, it's easy to classify the simples in $\mathcal{O}$, but much harder to describe them explicitly. The best we have done is to give an algorithm (which goes by the name Kazhdan-Lusztig theory) computing the Cartan matrix (it's actually somewhat finer information relating the simples to certain induced modules called Verma modules).