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, 8 insertions, 5 deletions

diff --git a/sections/semisimple-algebras.tex b/sections/semisimple-algebras.tex
@@ -459,11 +459,14 @@ order in \(Q\), where elements are ordered by their \emph{heights}.
 \end{definition}
 
 \begin{definition}
-  Given a basis \(\Sigma\) for \(\Delta\), there is a canonical partition
-  \(\Delta^+ \cup \Delta^- = \Delta\), where \(\Delta^+ = \{ \alpha \in \Delta
-  : \alpha \succeq 0 \}\) and \(\Delta^- = \{ \alpha \in \Delta : \alpha
-  \preceq 0 \}\). The elements of \(\Delta^+\) and \(\Delta^-\) are called
-  \emph{positive} and \emph{negative roots}, respectively.
+  Given a basis \(\Sigma\) for \(\Delta\), there is a canonical
+  partition\footnote{Notice that $\operatorname{ht}(\alpha) = 0$ if, and only
+  if $\alpha = 0$. Since $0$ is, by definition, not a root, the sets $\Delta^+$
+  and $\Delta^-$ account for all roots.} \(\Delta^+ \cup \Delta^- = \Delta\),
+  where \(\Delta^+ = \{ \alpha \in \Delta : \alpha \succ 0 \}\) and \(\Delta^-
+  = \{ \alpha \in \Delta : \alpha \prec 0 \}\). The elements of \(\Delta^+\)
+  and \(\Delta^-\) are called \emph{positive} and \emph{negative roots},
+  respectively.
 \end{definition}
 
 \begin{definition}