diff --git a/sections/semisimple-algebras.tex b/sections/semisimple-algebras.tex
@@ -19,7 +19,7 @@ restrictions we impose are twofold: restrictions on the algebras whose
representations we'll classify, and restrictions on the representations
themselves. First of all, we will work exclusively with finite-dimensional Lie
algebras over an algebraicly closed field \(K\) of characteristic \(0\). This
-is a restriction we will cary throught this notes. Moreover, as indicated by
+is a restriction we will cary throught these notes. Moreover, as indicated by
the title of this chapter, we will initially focus on the so called
\emph{semisimple} Lie algebras algebras\footnote{We will later relax this
restriction a bit in the next chapter.}. There are multiple equivalent ways to
@@ -246,7 +246,7 @@ equivalent to complete reducibility -- if, and only if \(H^1(\mathfrak{g},
%characterization of definition~\ref{def:semisimple-is-direct-sum}.
%
%To conclude this dubious attempt at a proof, we refer to a theorem by Hermann
-%Weyl, whose proof is beyond the scope of this notes as it requires calculating
+%Weyl, whose proof is beyond the scope of these notes as it requires calculating
%the Ricci curvature of \(G\) \footnote{The Ricci curvature is a tensor related
%to any given connection in a manifold. In this proof we're interested in the
%Ricci curvature of the Riemannian connection of \(\widetilde H\) under the
@@ -275,7 +275,7 @@ equivalent to complete reducibility -- if, and only if \(H^1(\mathfrak{g},
% group. We are done.
%\end{proof}
%
-%This results can be generalized to a certain extent by considering the exact
+%These results can be generalized to a certain extent by considering the exact
%sequence
%\begin{center}
% \begin{tikzcd}
@@ -316,7 +316,7 @@ equivalent to complete reducibility -- if, and only if \(H^1(\mathfrak{g},
% TODO: This shouldn't be considered underwelming! The primary results of this
% notes are concerned with irreducible representations of reducible Lie
% algebras
-This results can be generalized to a certain extent by considering the exact
+These results can be generalized to a certain extent by considering the exact
sequence
\begin{center}
\begin{tikzcd}