lie-algebras-and-their-representations

diff --git a/sections/semisimple-algebras.tex b/sections/semisimple-algebras.tex
@@ -1,10 +1,9 @@
 \chapter{Finite-Dimensional Irreducible Representations}
 
-% TODO: Write an introduction
-
-At the heart of our analysis of \(\mathfrak{sl}_2(K)\) and
-\(\mathfrak{sl}_3(K)\) was the decision to consider the eigenspace
-decomposition
+In this chapter we classify the finite-dimensional irreducible representations
+of a finite-dimensional semisimple Lie algebra \(\mathfrak{g}\) over \(K\). At
+the heart of our analysis of \(\mathfrak{sl}_2(K)\) and \(\mathfrak{sl}_3(K)\)
+was the decision to consider the eigenspace decomposition
 \begin{equation}\label{sym-diag}
   V = \bigoplus_\lambda V_\lambda
 \end{equation}
@@ -284,15 +283,21 @@ be starting become clear, so we will mostly omit technical details and proofs
 analogous to the ones on the previous sections. Further details can be found in
 appendix D of \cite{fulton-harris} and in \cite{humphreys}.
 
-% TODO: Write a transition
-
 \section{The Geometry of Roots and Weights}
 
-We begin our analysis by remarking that in both \(\mathfrak{sl}_2(K)\) and
-\(\mathfrak{sl}_3(K)\), the roots were symmetric about the origin and spanned
-all of \(\mathfrak{h}^*\). This turns out to be a general fact, which is a
-consequence of the non-degeneracy of the restriction of the Killing form to the
-Cartan subalgebra.
+We begin our analysis, as we did for \(\mathfrak{sl}_2(K)\) and
+\(\mathfrak{sl}_3(K)\), by investigating the set of roots of and weights of
+\(\mathfrak{g}\). Throughout chapter~\ref{ch:sl3} we've seen that the weights
+of any given finite-dimensional representation of \(\mathfrak{sl}_2(K)\) or
+\(\mathfrak{sl}_3(K)\) can only assume very rigit configurations. For instance,
+we've seen that the weights of any given representation are symmetric with
+respect to the origin. In this chapter we will generalize most results from
+chapter~\ref{ch:sl3} the rigidity of the geometry of the set of weights of a
+given representations.
+
+As for the affor mentioned result on the symmetry of roots, this turns out to
+be a general fact, which is a consequence of the non-degeneracy of the
+restriction of the Killing form to the Cartan subalgebra.
 
 \begin{proposition}\label{thm:weights-symmetric-span}
   The eigenvalues \(\alpha\) of the adjoint action of \(\mathfrak{h}\) in
@@ -583,7 +588,8 @@ Killing for to \(\mathfrak{h}\) we get a linear automorphism \(\mathfrak{h}
 \isoto \mathfrak{h}\). As it turns out, the automorphism
 \(\sigma\!\restriction_{\mathfrak{h}} : \mathfrak{h} \isoto \mathfrak{h}\) can
 be extended to an automorphism of Lie algebras \(\mathfrak{g} \isoto
-\mathfrak{g}\). Namely\dots
+\mathfrak{g}\). This translates into the following results, which we do not
+prove -- but see \cite[sec.~14.3]{humphreys}.
 
 \begin{proposition}\label{thm:weyl-group-action}
   Given \(\alpha \in \Delta^+\), let\footnote{Notice that since $\mathfrak{g}$
@@ -610,12 +616,16 @@ be extended to an automorphism of Lie algebras \(\mathfrak{g} \isoto
   \(\mathfrak{h}\) is independent of any choices.
 \end{note}
 
-See \cite[sec.~14.3]{humphreys} for a complete proof. Now the only thing we are
-missing for a complete classification is an existence and uniqueness theorem
-analogous to theorem~\ref{thm:sl2-exist-unique} and
-theorem~\ref{thm:sl3-existence-uniqueness}.
-
-% TODO: Write a transition
+We should point out that the results in this section regarding the geometry
+roots and weights are only the begining of a well develop axiomatic theory of
+the so called \emph{root systems}, which was used by Cartan in the early 20th
+century to classify all finite-dimensional simple complex Lie algebras in terms
+of Dynking diagrams. This and much more can be found in \cite[III]{humphreys}
+and \cite[ch.~21]{fulton-harris}. Having found all of the weights of \(V\), the
+only thing we are missing for a complete classification is an existence and
+uniqueness theorem analogous to theorem~\ref{thm:sl2-exist-unique} and
+theorem~\ref{thm:sl3-existence-uniqueness}. This will be the focus of the next
+section.
 
 \section{Verma Modules \& the Highest Weight Theorem}
 
@@ -881,7 +891,7 @@ are really interested in is\dots
 \end{corollary}
 
 \begin{proof}
-  Let \(V = \sfrac{M(\lambda)}{N(\lambda)}\). It suffices to show that its
+  Let \(V = \mfrac{M(\lambda)}{N(\lambda)}\). It suffices to show that its
   highest weight is \(\lambda\). We have already seen that \(v^+ \in
   M(\lambda)_\lambda\) is a highest weight vector. Now since \(v\) lies outside
   of the maximal subrepresentation of \(M(\lambda)\), the projection \(v^+ +
@@ -899,6 +909,22 @@ are really interested in is\dots
   weight \(\mu\) of \(M(\lambda)\) which is higher than \(\lambda\).
 \end{proof}
 
-% TODO: Write a conclusion
-
-
+We should point out that proposition~\ref{thm:verma-is-finite-dim} fails for
+nondominants \(\lambda \in P\). While \(\lambda\) is always a maximal weight of
+\(M(\lambda)\), one can show show that if \(\lambda \in P\) is not dominant
+then \(N(\lambda) = 0\) and \(M(\lambda)\) is irreducible. For instance, if
+\(\mathfrak{g} = \mathfrak{sl}_2(K)\) and \(\lambda = -2\) then the action of
+\(\mathfrak{g}\) in \(M(\lambda)\) is given by
+\begin{center}
+  \begin{tikzcd}
+    \cdots \arrow[bend left=60]{r}{-20}
+    & M(\lambda)_{-8}  \arrow[bend left=60]{r}{-12} \arrow[bend left=60]{l}{1}
+    & M(\lambda)_{-6}  \arrow[bend left=60]{r}{-6}  \arrow[bend left=60]{l}{1}
+    & M(\lambda)_{-4}  \arrow[bend left=60]{r}{-2}  \arrow[bend left=60]{l}{1}
+    & M(\lambda)_{-2}                               \arrow[bend left=60]{l}{1}
+  \end{tikzcd},
+\end{center}
+so we can see that \(M(-2)\) has no proper subrepresentations. Verma modules
+can thus serve as examples of infinite-dimensional irreducible representations.
+Our next question is: what are \emph{all} the infinite-dimensional irreducible
+\(\mathfrak{g}\)-modules?