lie-algebras-and-their-representations

Source code for my notes on representations of semisimple Lie algebras and Olivier Mathieu's classification of simple weight modules

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1 file changed, 7 insertions, 6 deletions

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -530,12 +530,13 @@ semisimple and reductive algebras by modding out by certain ideals, known as
 \end{proposition}
 
 We have seen in Example~\ref{ex:inclusion-alg-in-lie-alg} that we can pass from
-associative algebras to Lie algebras using the functor \(\operatorname{Lie} :
-K\text{-}\mathbf{Alg} \to K\text{-}\mathbf{LieAlg}\) that takes an algebra
-\(A\) to the Lie algebra \(A\) with bracket given by commutators. We can also
-go the other direction by embedding a Lie algebra \(\mathfrak{g}\) in an
-associative algebra, known as \emph{the universal enveloping algebra of
-\(\mathfrak{g}\)}.
+an associative algebra \(A\) to a Lie algebra by taking its bracket as the
+commutator \([a, b] = ab - ba\). We should also not that any homomorphism of
+\(K\)-algebras \(f : A \to B\) preserves commutators, so that \(f\) is also a
+homomorphism of Lie algebras. Hence we have a functor \(\operatorname{Lie} :
+K\text{-}\mathbf{Alg} \to K\text{-}\mathbf{LieAlg}\). We can also go the other
+direction by embedding a Lie algebra \(\mathfrak{g}\) in an associative
+algebra, known as \emph{the universal enveloping algebra of \(\mathfrak{g}\)}.
 
 \begin{definition}
   Let \(\mathfrak{g}\) be a Lie algebra and \(T \mathfrak{g} = \bigoplus_n