diff --git a/sections/sl2-sl3.tex b/sections/sl2-sl3.tex
@@ -66,7 +66,7 @@ and the eigenvalues of \(h\) all have the form \(\lambda - 2 k\) for some
Even more so, if \(a = \max \{ k \in \mathbb{Z} : V_{\lambda - 2 k} \ne 0 \}\) and
\(b = \min \{ k \in \mathbb{Z} : V_{\lambda - 2 k} \ne 0 \}\) we can see that
\[
- \bigoplus_{\substack{k \in \mathbb{Z} \\ a \le n \le b}} V_{\lambda - 2 k}
+ \bigoplus_{\substack{k \in \mathbb{Z} \\ a \le k \le b}} V_{\lambda - 2 k}
\]
is also an \(\mathfrak{sl}_2(K)\)-invariant subspace, so that the eigenvalues
of \(h\) form an unbroken string
@@ -85,9 +85,9 @@ V_\lambda\) and consider the set \(\{v, f v, f^2 v, \ldots\}\).
action of \(\mathfrak{sl}_2(K)\) on \(V\) is given by the formulas
\begin{equation}\label{eq:irr-rep-of-sl2}
\begin{aligned}
- f^k v & \overset{e}{\mapsto} k(n + 1 - k) f^{k - 1} v
+ f^k v & \overset{e}{\mapsto} k(\lambda + 1 - k) f^{k - 1} v
& f^k v & \overset{f}{\mapsto} f^{k + 1} v
- & f^k v & \overset{h}{\mapsto} (n - 2 k) f^k v
+ & f^k v & \overset{h}{\mapsto} (\lambda - 2 k) f^k v
\end{aligned}
\end{equation}
\end{proposition}
@@ -203,10 +203,10 @@ In particular, if the eigenvalues of \(V\) all have the same parity -- i.e.
they are either all even integers or all odd integers -- and the dimension of
each eigenspace is no greater than \(1\) then \(V\) must be irreducible, for if
\(U, W \subset V\) are subrepresentations with \(V = W \oplus U\) then either
-\(W_\lambda = 0\) for all \(\lambda\) or \(U_\lambda = 0\) for all \(\lambda\).
-To conclude our analysis all it is left is to show that for each \(n\) there is
-some finite-dimensional irreducible \(V\) whose highest weight is \(\lambda\).
-Surprisingly, we have already encountered such a \(V\).
+\(W_\lambda = 0\) for all \(\lambda\) or \(U_\lambda = 0\) for all \(\lambda
+\in \mathfrak{h}^*\). To conclude our analysis all it is left is to show that
+for each \(n\) there is some finite-dimensional irreducible \(V\) whose highest
+weight is \(\lambda\). Surprisingly, we have already encountered such a \(V\).
\begin{theorem}\label{thm:sl2-exist-unique}
For each \(n \ge 0\) there exists a unique irreducible representation of
@@ -536,9 +536,9 @@ Visually,
In general, we find\dots
\begin{proposition}
- Given \(i < j\), the subalgebra \(\mathfrak{s}_{\alpha_i - \alpha_j} = K
- \langle E_{i j}, E_{j i}, [E_{i j}, E_{j i}] \rangle\) is isomorphic to
- \(\mathfrak{sl}_2(K)\). In addition, given weight \(\lambda \in
+ Given \(i < j\), the subalgebra \(\mathfrak{s}_{i j} = K \langle E_{i j},
+ E_{j i}, [E_{i j}, E_{j i}] \rangle\) is isomorphic to
+ \(\mathfrak{sl}_2(K)\). In addition, given a weight \(\lambda \in
\mathfrak{h}^*\) of \(V\), the space
\[
W = \bigoplus_{k \in \mathbb{Z}} V_{\lambda - k (\alpha_i - \alpha_j)}