diff --git a/sections/semisimple-algebras.tex b/sections/semisimple-algebras.tex
@@ -119,7 +119,7 @@ What is simultaneous diagonalization all about then?
\end{definition}
\begin{proposition}
- Given a \emph{finite-dimensional} vector space \(V\), A set of diagonalizable
+ Given a \emph{finite-dimensional} vector space \(V\), a set of diagonalizable
operators \(V \to V\) is simultaneously diagonalizable if, and only if all of
its elements commute with one another.
\end{proposition}
@@ -158,8 +158,8 @@ words\dots
\begin{proposition}\label{thm:preservation-jordan-form}
Let \(V\) be a finite-dimensional representation of \(\mathfrak{g}\) and \(X
\in \mathfrak{g}\). Denote by \(X\!\restriction_V\) the action of \(X\) on
- \(V\). Then \(X_s\!\restriction_V = (X\!\restriction)_s\) and
- \(X_n\!\restriction_V = (X\!\restriction)_n\).
+ \(V\). Then \(X_s\!\restriction_V = (X\!\restriction_V)_s\) and
+ \(X_n\!\restriction_V = (X\!\restriction_V)_n\).
\end{proposition}
This last result is known as \emph{the preservation of the Jordan form}, and a
@@ -291,9 +291,9 @@ are symmetric with respect to the origin. In this chapter we will generalize
most results from chapter~\ref{ch:sl3} regarding the rigidity of the geometry
of the set of weights of a given representations.
-As for the afford 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.
+As for the aforementioned 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 roots \(\alpha\) of \(\mathfrak{g}\) are symmetrical about the origin --
@@ -353,21 +353,22 @@ each \(H \in \mathfrak{h}\) and \(v \in V_\lambda\) we find
= X (H v) + [H, X] v
= (\lambda + \alpha)(H) \cdot X v
\]
-so that \(X\) carries \(v\) to \(V_{\lambda + \alpha}\). We have encountered
-this formula twice in these notes: again, we find \(\mathfrak{g}_\alpha\)
-\emph{acts on \(V\) by translating vectors between eigenspaces}. In particular,
-if we denote by \(\Delta\) the set of all roots of \(\mathfrak{g}\) then\dots
+
+Thus \(X\) sends \(v\) to \(V_{\lambda + \alpha}\). We have encountered this
+formula twice in these notes: again, we find \(\mathfrak{g}_\alpha\) \emph{acts
+on \(V\) by translating vectors between eigenspaces}. In particular, if we
+denote by \(\Delta\) the set of all roots of \(\mathfrak{g}\) then\dots
\begin{theorem}\label{thm:weights-congruent-mod-root}
The weights of an irreducible representation \(V\) of \(\mathfrak{g}\) are
- all congruent module the root lattice \(Q = \mathbb{Z} \Delta\) of \(\mathfrak{g}\).
+ all congruent modulo the root lattice \(Q = \mathbb{Z} \Delta\) of \(\mathfrak{g}\).
In other words, all weights of \(V\) lie in the same \(Q\)-coset
\(t \in \mfrac{\mathfrak{h}^*}{Q}\).
\end{theorem}
-Again, we may leverage our knowledge of \(\mathfrak{sl}_2(K)\) to obtain further
-restrictions on the geometry of the space of weights of \(V\). Namely, such as
-in the case of \(\mathfrak{sl}_3(K)\) we show\dots
+Again, we may leverage our knowledge of \(\mathfrak{sl}_2(K)\) to obtain
+further restrictions on the geometry of the space of weights of \(V\). Namely,
+as in the case of \(\mathfrak{sl}_3(K)\) we show\dots
\begin{proposition}\label{thm:distinguished-subalgebra}
Given a root \(\alpha\) of \(\mathfrak{g}\) the subspace
@@ -404,7 +405,7 @@ The elements \(E_\alpha, F_\alpha \in \mathfrak{g}\) are not uniquely
determined by this condition, but \(H_\alpha\) is. As promised, the second
statement of corollary~\ref{thm:distinguished-subalg-rep} imposes strong
restrictions on the weights of \(V\). Namely, if \(\lambda\) is a weight,
-\(\lambda(H_\alpha)\) is an eigenvalue of \(h\) in some representation of
+\(\lambda(H_\alpha)\) is an eigenvalue of \(h\) on some representation of
\(\mathfrak{sl}_2(K)\), so it must be an integer. In other words\dots
\begin{definition}\label{def:weight-lattice}
@@ -783,14 +784,8 @@ Moreover, we find\dots
h\) and \(\mathfrak{b} = K e \oplus K h\). If \(\lambda \in
\mathfrak{h}^*\) is the map \(h \mapsto 2\) then \(M(\lambda) =
\bigoplus_{k \ge 0} K f^k v^+\), and the action of \(\mathfrak{sl}_2(K)\) on
- \(M(\lambda)\) is given by
- \begin{align*}
- f^k v^+ & \overset{e}{\mapsto} (2 - k (k + 1)) f^{k - 1} v^+ &
- f^k v^+ & \overset{f}{\mapsto} f^{k + 1} v^+ &
- f^k v^+ & \overset{h}{\mapsto} - 2 (k - 1) f^k v^+ &
- \end{align*}
-
- In the language of the diagrams used in chapter~\ref{ch:sl3}, we write
+ \(M(\lambda)\) is given by the formulas in (\ref{eq:sl2-verma-formulas}).
+ Visually,
\begin{center}
\begin{tikzcd}
\cdots \arrow[bend left=60]{r}{-10}
@@ -802,9 +797,17 @@ Moreover, we find\dots
\end{tikzcd}
\end{center}
where \(M(\lambda)_{2 - 2 k} = K f^k v\). Here the top arrows represent the
- action of \(e\) and the bottom arrows represent the action of \(f\). In this
- case, unlike we have see in chapter~\ref{ch:sl3}, the string of weight spaces
- to left of the diagram is infinite.
+ action of \(e\) and the bottom arrows represent the action of \(f\). The
+ scalars labeling each arrow indicate to which multiple of \(f^{k \pm 1} v\)
+ the elements \(e\) and \(f\) send \(f^k v\). The string of weight spaces to
+ the left of the diagram is infinite.
+ \begin{equation}\label{eq:sl2-verma-formulas}
+ \begin{aligned}
+ f^k v^+ & \overset{e}{\mapsto} (2 - k (k + 1)) f^{k - 1} v^+ &
+ f^k v^+ & \overset{f}{\mapsto} f^{k + 1} v^+ &
+ f^k v^+ & \overset{h}{\mapsto} (2 - 2k) f^k v^+ &
+ \end{aligned}
+ \end{equation}
\end{example}
What's interesting to us about all this is that we've just constructed a
@@ -947,7 +950,7 @@ action of \(\mathfrak{g}\) on \(M(\lambda)\) is given by
& 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?
+so we can see that \(M(\lambda)\) 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?