diff --git a/sections/sl2-sl3.tex b/sections/sl2-sl3.tex
@@ -564,17 +564,41 @@ In general, we find\dots
\end{proposition}
\begin{proof}
- For the first claim, it suffices to notice the map
+ In effect, if \(i \ne k \ne j\) then \(\mathfrak{s}_{i j}\) is the subalgebra
+ of matrices whose \(k\)th row and \(k\)th collumn are nil. For instance, if
+ \(i = 1\) and \(j = 3\) then
+ \[
+ \mathfrak{s}_{1 3}
+ = \begin{pmatrix} K & 0 & K \\ 0 & 0 & 0 \\ K & 0 & K \end{pmatrix}
+ \cap \mathfrak{sl}_3(K)
+ \]
+
+ In this case, the map
\begin{align*}
- \mathfrak{sl}_2(K) & \to \mathfrak{s}_{i j} \\
- e & \mapsto E_{i j} \\
- f & \mapsto E_{j i} \\
- h & \mapsto [E_{i j}, E_{j i}]
+ \mathfrak{s}_{1 3} & \to \mathfrak{sl}_2(K) \\
+ \begin{pmatrix} a & 0 & b \\ 0 & 0 & 0 \\ c & 0 & -a \end{pmatrix}
+ & \mapsto
+ \begin{pmatrix}
+ a & \tm{topA}{0} & b \\
+ \tm{leftA}{0} & 0 & \tm{rightA}{0} \\
+ c & \tm{bottomA}{0} & -a
+ \end{pmatrix}
+ = \begin{pmatrix} a & b \\ c & -a \end{pmatrix}
+ \DrawVLine[black, thick, opacity=0.5]{topA}{bottomA}
+ \DrawHLine[black, thick, opacity=0.5]{leftA}{rightA}
\end{align*}
- is an isomorphism.
+ is an isomorphism of Lie algebras. In general, the map
+ \begin{align*}
+ \mathfrak{s}_{i j} & \to \mathfrak{sl}_2(K) \\
+ E_{i j} & \mapsto e \\
+ E_{j i} & \mapsto f \\
+ [E_{i j}, E_{j i}] & \mapsto h
+ \end{align*}
+ which ``erases the \(k\)th row and the \(k\)th collumn'' of a matrix is an
+ isomorphism.
To see that \(W\) is invariant under the action of \(\mathfrak{s}_{i j}\), it
- suffices to notice \(E_{i j}\) and \(E_{j i}\) map \(v \in V_{\lambda + k
+ suffices to notice \(E_{i j}\) and \(E_{j i}\) map \(v \in V_{\lambda - k
(\alpha_i - \alpha_j)}\) to \(E_{i j} v \in V_{\lambda - (k - 1) (\alpha_i -
\alpha_j)}\) and \(E_{j i} v \in V_{\lambda - (k + 1) (\alpha_i -
\alpha_j)}\). Moreover,
@@ -596,9 +620,6 @@ In general, we find\dots
a contradiction.
\end{proof}
-% TODO: Note that the subalgebra s_ij is obtained by "excluding" a given row
-% and a given column from the matrices in sl3
-
As a first consequence of this, we show\dots
\begin{definition}