lie-algebras-and-their-representations

diff --git a/references.bib b/references.bib
@@ -218,3 +218,14 @@
   year = {2020},
   copyright = {Creative Commons Zero v1.0 Universal}
 }
+
+@article{nilsson,
+title = {$\mathcal{U}(\mathfrak{g})$-free modules and coherent families},
+journal = {Journal of Pure and Applied Algebra},
+volume = {220},
+number = {4},
+pages = {1475-1488},
+year = {2016},
+doi = {https://doi.org/10.1016/j.jpaa.2015.09.013},
+author = {Jonathan Nilsson},
+}

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -17,32 +17,41 @@ classification of completely reducible representations. Our strategy is, once
 again, to classify the irreducible representations.
 
 % TODO: Add references to the motivation
+% TODO: Point out that U(g) is always a domain in the introduction
 Secondly, and this is more important, we now consider
 \emph{infinite-dimensional} representations too. The motivation behind looking
 at infinite-dimensional modules was already explained in the introduction, but
-this introduces numerous complications to our analysis. For instance, \(h\)
-acts freely\footnote{To see this, it suffices to notice that the Cartan
-subalgebra has a complementary subalgebra in $\mathfrak{sl}_2(K)$ and apply the
-Poincaré-Birkoff-Witt theorem.} in \(\mathcal{U}(\mathfrak{sl}_2(K))\) -- i.e.
-\(\mathcal{U}(\mathfrak{sl}_2(K))\) is a free \(\mathfrak{h}\)-module for
-\(\mathfrak{h} = K h\). In particular, the weight spaces decomposition
+this introduces numerous complications to our analysis. For example, if
+\(\mathcal{U}(\mathfrak{g})\) is the regular \(\mathfrak{g}\)-module then
+\(\mathcal{U}(\mathfrak{g})_\lambda = 0\) for all \(\lambda \in
+\mathfrak{h}^*\). This follows from the fact that \(\mathcal{U}(\mathfrak{g})\)
+has no zero divisors: given \(u \in \mathcal{U}(\mathfrak{g})\), \((H -
+\lambda(H)) u = 0\) for some nonzero \(H \in \mathfrak{h}\) implies \(u = 0\).
+In particular,
 \[
-  \mathcal{U}(\mathfrak{sl}_2(K))
-  = \bigoplus_\lambda \mathcal{U}(\mathfrak{sl}_2(K))_\lambda
+  \bigoplus_\lambda \mathcal{U}(\mathfrak{g})_\lambda
+  = 0
+  \subsetneq \mathcal{U}(\mathfrak{g})
 \]
-\emph{does not} hold for the regular \(\mathfrak{sl}_2(K)\)-module
-\(\mathcal{U}(\mathfrak{sl}_2(K))\), for if % TODOOOOOOOO
+and the weight space decomposition -- i.e.
+corollary~\ref{thm:finite-dim-is-weight-mod} -- fails for
+\(\mathcal{U}(\mathfrak{g})\).
 
-% TODO: Comment on Nilson's work with h-free modules?
 Indeed, our proof of corollary~\ref{thm:finite-dim-is-weight-mod} relied
 heavily in the simultaneous diagonalization of commuting operators in a
-finite-dimensional space, which is usually framed as statement about matrices.
-Even if we restrict ourselves to the irreducible case, there is still a diverse
-spectrum of conterexamples to corollary~\ref{thm:finite-dim-is-weight-mod} in
-the infinite-dimensional setting. For instace, see Dimitar's construction of
-the so called \emph{exponential tensor \(\mathfrak{sl}_n(K)\)-modules} in
-\cite{dimitar-exp}. Since the weight spaces decomposition \(V =
-\bigoplus_\lambda V_\lambda\) was perhaps the single most instrumental
+finite-dimensional space. Even if we restrict ourselves to irreducible modules,
+there is still a diverse spectrum of conterexamples to
+corollary~\ref{thm:finite-dim-is-weight-mod} in the infinite-dimensional
+setting. For instace, any representation \(V\) of \(\mathfrak{g}\) whose
+restriction to \(\mathfrak{h}\) is a free module satisfies \(V_\lambda = 0\)
+for all \(\lambda\) as in the previous example. These are called
+\(\mathfrak{h}\)-free representations, and rank \(1\) irreducible
+\(\mathfrak{h}\)-free \(\mathfrak{sp}_{2 n}(K)\)-modules where first classified
+by Nilsson in \cite{nilsson}. Dimitar's construction of the so called
+\emph{exponential tensor \(\mathfrak{sl}_n(K)\)-modules} in \cite{dimitar-exp}
+is also an interesting class of counterexamples.
+
+Since the weight spaces decomposition was perhaps the single most instrumental
 ingrediant of our previous analysis, it is only natural to restrict ourselves
 to the case it holds. This brings us to the following definition.