lie-algebras-and-their-representations

diff --git a/sections/complete-reducibility.tex b/sections/complete-reducibility.tex
@@ -614,8 +614,8 @@ obstructions to complete reducibility. Explicitly\dots
 
 \begin{theorem}
   Given \(\mathfrak{g}\)-modules \(N\) and \(L\), there is a one-to-one
-  correspondence between elements of \(H^1(\mathfrak{g}, \operatorname{Hom}(N,
-  L))\) and isomorphism classes of short exact sequences
+  correspondence between elements of \(H^1(\mathfrak{g}, \operatorname{Hom}(L,
+  N))\) and isomorphism classes of short exact sequences
   \begin{center}
     \begin{tikzcd}
       0 \rar & N \rar & M \rar & L \rar & 0
@@ -647,7 +647,7 @@ for further details.
 We will use the previous result implicitly in our proof, but we will not prove
 it in its full force. Namely, we will show that if \(\mathfrak{g}\) is
 semisimple then \(H^1(\mathfrak{g}, M) = 0\) for all finite-dimensional \(M\),
-and that the fact that \(H^1(\mathfrak{g}, \operatorname{Hom}(N, L)) = 0\) for
+and that the fact that \(H^1(\mathfrak{g}, \operatorname{Hom}(L, N)) = 0\) for
 all finite-dimensional \(N\) and \(L\) implies complete reducibility. To that
 end, we introduce a distinguished element of \(\mathcal{U}(\mathfrak{g})\),
 known as \emph{the Casimir element of a \(\mathfrak{g}\)-module}.