lie-algebras-and-their-representations

diff --git a/references.bib b/references.bib
@@ -192,3 +192,11 @@
   year={2022},
   url={https://perso.pages.math.cnrs.fr/users/gabriel.ribeiro/assets/files/main.pdf}
 }
+
+@book{symplectic-physics,
+   title =     {Symplectic techniques in physics},
+   author =    {Victor Guillemin, Shlomo Sternberg},
+   publisher = {Cambridge University Press},
+   year =      {1984},
+   edition =   {First Edition},
+}

diff --git a/sections/semisimple-algebras.tex b/sections/semisimple-algebras.tex
@@ -401,14 +401,15 @@ can computed very concretely by considering a canonical acyclic resolution
   \end{tikzcd}
 \end{center}
 of the trivial representation \(K\), which provides an explicit construction of
-the cohomology groups -- see \cite[sec. 9]{lie-groups-serganova-student} 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
-\(H^1(\mathfrak{g}, V) = 0\) for all finite-dimensional \(V\), and that the
-fact that \(H^1(\mathfrak{g}, \operatorname{Hom}(W, U)) = 0\) for all
-finite-dimensional \(W\) and \(U\) implies complete reducibility. To that end,
-we introduce a distinguished element of \(\mathcal{U}(\mathfrak{g})\), known as
-\emph{the Casimir element of a representation}.
+the cohomology groups -- see \cite[sec.~9]{lie-groups-serganova-student} or
+\cite[sec.~24]{symplectic-physics} 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 \(H^1(\mathfrak{g}, V) = 0\) for all
+finite-dimensional \(V\), and that the fact that \(H^1(\mathfrak{g},
+\operatorname{Hom}(W, U)) = 0\) for all finite-dimensional \(W\) and \(U\)
+implies complete reducibility. To that end, we introduce a distinguished
+element of \(\mathcal{U}(\mathfrak{g})\), known as \emph{the Casimir element of
+a representation}.
 
 \begin{definition}\label{def:casimir-element}
   Let \(V\) be a finite-dimensional representation of \(\mathfrak{g}\).
@@ -699,8 +700,8 @@ non-trivial finite-dimensional irreducible \(V\), \(i > 0\). For \(K =
 \mathbb{C}\), the Lie algebra cohomology groups of an algebra \(\mathfrak{g} =
 \mathbb{C} \otimes \operatorname{Lie}(G)\) are intimately related with the
 topological cohomologies -- i.e. singular cohomology, de Rham cohomology, etc.
--- of \(G\). We refer the reader to \cite{cohomologies-lie} for further
-details.
+-- of \(G\). We refer the reader to \cite{cohomologies-lie} and
+\cite[sec.~24]{symplectic-physics} for further details.
 
 Complete reducibility can be generalized for arbitrary -- not necessarily
 semisimple -- \(\mathfrak{g}\), to a certain extent, by considering the exact