lie-algebras-and-their-representations

diff --git a/references.bib b/references.bib
@@ -209,3 +209,12 @@
    series =    {London Mathematical Society Student Texts},
    edition =   {Second Edition},
 }
+
+@misc{dimitar-exp,
+  doi = {10.48550/ARXIV.2011.09975},
+  author = {Dimitar Grantcharov, Khoa Nguyen},
+  title = {Exponentiation and Fourier transform of tensor modules of $\mathfrak{sl} (n+1)$},
+  publisher = {arXiv},
+  year = {2020},
+  copyright = {Creative Commons Zero v1.0 Universal}
+}

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -1,5 +1,51 @@
 \chapter{Irreducible Weight Modules}\label{ch:mathieu}
 
+In this chapter we'll expand our results on finite-dimensional irreducible
+representations of semisimple Lie algebras by generalizing them on multiple
+directions. First, we will now consider reductive Lie algebras, which means we
+can no longer take complete reducibility for granted. Namely, we've seen that
+if \(\mathfrak{g}\) is \emph{not} semisimple there must be some
+\(\mathfrak{g}\)-module which is not the direct sum of irreducible
+representations.
+
+Nevertheless, completely reducible representations are a \emph{very} large
+class of \(\mathfrak{g}\)-modules, and understanding them can still give us a
+lot of information regarding our algebre and the category of its
+representations -- granted, not \emph{all} of the information as in the
+semisimple case. For this reason, we will focus exclusively on the
+classification of completely reducible representations. Our strategy is, once
+again, to classify the irreducible representations.
+
+% TODO: Add references to the motivation
+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
+\[
+  \mathcal{U}(\mathfrak{sl}_2(K))
+  = \bigoplus_\lambda \mathcal{U}(\mathfrak{sl}_2(K))_\lambda
+\]
+\emph{does not} hold for the regular \(\mathfrak{sl}_2(K)\)-module
+\(\mathcal{U}(\mathfrak{sl}_2(K))\), for if % TODOOOOOOOO
+
+% 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
+ingrediant of our previous analysis, it is only natural to restrict ourselves
+to the case it holds. This brings us to the following definition.
+
 \begin{definition}
   A representation \(V\) of \(\mathfrak{g}\) is called a \emph{weight
   \(\mathfrak{g}\)-module} if \(V = \bigoplus_{\lambda \in \mathfrak{h}^*}
@@ -58,9 +104,8 @@
   \left(\mfrac{V}{W}\right)_\lambda\) is surjective.
 \end{example}
 
-% TODO: Add an example of a module wich is NOT a weight module
-
 % TODOO: Prove this? Most likely not!
+% TODOO: Move this to somewhere else? Its kind of an akward place for this
 \begin{proposition}\label{thm:centralizer-multiplicity}
   Let \(V\) be a completely reducible weight \(\mathfrak{g}\)-module. Then
   \(V_\lambda\) is a semisimple
@@ -73,6 +118,9 @@
   for any \(\lambda \in \operatorname{supp} V\).
 \end{proposition}
 
+A particularly well behaved class of examples are the so called
+\emph{admissible} weight modules.
+
 \begin{definition}
   A weight \(\mathfrak{g}\)-module is called \emph{admissible} if \(\dim
   V_\lambda\) is bounded. The lowest upper bound for \(\dim V_\lambda\) is
@@ -88,13 +136,14 @@
   x^{-1}]_{2 k} = K x^k\) and \(K[x, x^{-1}]_\lambda = 0\) for any \(\lambda
   \notin 2 \mathbb{Z}\), so that \(K[x, x^{-1}] = \bigoplus_{k \in \mathbb{Z}}
   K x^k\) is a degree \(1\) admissible weight \(\mathfrak{sl}_2(K)\)-module. It
-  follows from example~\ref{ex:submod-is-weight-mod} that any non-zero
-  subrepresentation \(W \subset K[x, x^{-1}]\) must contain a monomial \(x^k\).
-  But since the operators \(-\frac{\mathrm{d}}{\mathrm{d}x} + \frac{x^{-1}}{2},
-  x^2 \frac{\mathrm{d}}{\mathrm{d}x} + \frac{x}{2} : K[x, x^{-1}] \to K[x,
-  x^{-1}]\) are both injective, this implies all other monomials can be found
-  in \(W\) by successively applaying \(f\) and \(e\). Hence \(W = K[x,
-  x^{-1}]\) and \(K[x, x^{-1}]\) is an irreducible representation.
+  follows from the remark at the end of example~\ref{ex:submod-is-weight-mod}
+  that any non-zero subrepresentation \(W \subset K[x, x^{-1}]\) must contain a
+  monomial \(x^k\). But since the operators \(-\frac{\mathrm{d}}{\mathrm{d}x} +
+  \frac{x^{-1}}{2}, x^2 \frac{\mathrm{d}}{\mathrm{d}x} + \frac{x}{2} : K[x,
+  x^{-1}] \to K[x, x^{-1}]\) are both injective, this implies all other
+  monomials can be found in \(W\) by successively applaying \(f\) and \(e\).
+  Hence \(W = K[x, x^{-1}]\) and \(K[x, x^{-1}]\) is an irreducible
+  representation.
   \begin{align}\label{eq:laurent-polynomials-cusp-mod}
     f \cdot p
     & = \left(- \frac{\mathrm{d}}{\mathrm{d}x} + \frac{x^{-1}}{2} \right) p &
@@ -105,8 +154,21 @@
   \end{align}
 \end{example}
 
-% TODO: Point out supp_ess K[x^+-1] is 2Z, which is zariski dense
-% This proof is very technical, I don't think its worth including it
+Notice that the support of \(K[x, x^{-1}]\) is the trivial \(2
+\mathbb{Z}\)-coset \(0 + 2 \mathbb{Z}\). This is representative of the general
+behavious in the following sense: if \(V\) is an irreducible weight
+\(\mathfrak{g}\)-module, since \(\bigoplus_{\alpha \in Q} V_{\lambda +
+\alpha}\) is stable under the action of \(\mathfrak{g}\) for all \(\lambda \in
+\mathfrak{h}^*\), \(\bigoplus_{\alpha \in Q} V_{\lambda + \alpha}\) is either
+\(0\) or all \(V\). In other words, the support of an irreducible weight module
+is allways contained in a single \(Q\)-coset.
+
+However, the behaviour of \(K[x, x^{-1}]\) deviates from that of an arbitrary
+admissible representation in the sence its essential support is precisely the
+entire \(Q\)-coset it enhabits -- i.e.
+\(\operatorname{supp}_{\operatorname{ess}} K[x, x^{-1}] = 2 \mathbb{Z}\). This
+isn't always the case. Nevertheless, in general we find\dots
+
 \begin{proposition}
   Let \(V\) be an infinite-dimensional admissible representation of
   \(\mathfrak{g}\). The essential support
@@ -114,6 +176,12 @@
   \(\mathfrak{h}^*\).
 \end{proposition}
 
+Again, there is plenty of examples of completely reducible modules which are
+\emph{not} weight modules. Nevertheless, weight modules constitute a large
+class of representations and understanding them can give us a lot of insight
+into the general case. Our goal is now classifying all irreducible weight
+\(\mathfrak{g}\)-modules for some fixed reductive Lie algebra \(\mathfrak{g}\).
+
 \begin{definition}
   A subalgebra \(\mathfrak{p} \subset \mathfrak{g}\) is called \emph{parabolic}
   if \(\mathfrak{b} \subset \mathfrak{p}\).