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

Diffstat

1 file changed, 9 insertions, 9 deletions

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -521,15 +521,15 @@ Notice there is a canonical homomorphism \(\mathfrak{g} \to
   \end{tikzcd}
 \end{center}
 
-Given \(X, Y \in \mathfrak{g}\), we denote their images under the inclusion
-\(\mathfrak{g} \to T \mathfrak{g}\) simply by \(X\) and \(Y\) and we write \(X
-\cdot Y\) for \((X \otimes Y) + I\). This notation suggests the map
-\(\mathfrak{g} \to \mathcal{U}(\mathfrak{g})\) is injective, but at this point
-this is not at all clear -- since the projection \(T \mathfrak{g} \to
-\mathcal{U}(\mathfrak{g})\) is not injective. However, we will soon see this is
-the case. Intuitively, \(\mathcal{U}(\mathfrak{g})\) is the smallest
-associative \(K\)-algebra containing \(\mathfrak{g}\) as a Lie subalgebra. In
-practice this means\dots
+Given \(X_1, \ldots, X_n \in \mathfrak{g}\), we denote the image of \(X_i\)
+under the inclusion \(\mathfrak{g} \to T \mathfrak{g}\) simply by \(X_i\) and
+we write \(X_1 \cdots X_n\) for \((X_1 \otimes \cdots \otimes X_n) + I\). This
+notation suggests the map \(\mathfrak{g} \to \mathcal{U}(\mathfrak{g})\) is
+injective, but at this point this is not at all clear -- since the projection
+\(T \mathfrak{g} \to \mathcal{U}(\mathfrak{g})\) is not injective. However, we
+will soon see this is the case. Intuitively, \(\mathcal{U}(\mathfrak{g})\) is
+the smallest associative \(K\)-algebra containing \(\mathfrak{g}\) as a Lie
+subalgebra. In practice this means\dots
 
 \begin{proposition}\label{thm:universal-env-uni-prop}
   Let \(\mathfrak{g}\) be a Lie algebra and \(A\) be an associative