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, 6 deletions

diff --git a/sections/complete-reducibility.tex b/sections/complete-reducibility.tex
@@ -281,14 +281,16 @@ to introduce some basic tools which will come in handy later on, known as\dots
   \]
 \end{definition}
 
-% TODO: Note that antisymmetric matrices are the infinitesimal version of
-% ortogonal matrices
 \begin{note}
   The etymology of the term \emph{invariant form} comes from group
-  representation theory. If \(G \subset \operatorname{GL}(V)\) is a group of
-  linear automorphisms of a \(K\)-vector space \(V\), a bilinear form \(B : V
-  \times V \to K\) is called \(G\)-invariant if each \(g \in G\) is an
-  orthogonal operator with respect to the form \(B\).
+  representation theory. Namely, given a linear action of a group \(G\) on a
+  vector space \(V\) equipped with a bilinear form \(B\), \(B\) is called
+  \(G\)-invariant if all \(g \in G\) act via \(B\)-orthogonal operators. The
+  condition of \(\mathfrak{g}\)-invariance can thus be though-of as an
+  \emph{infinitesimal approximation} of the notion of a \(G\)-invariant form.
+  Indeed \(\operatorname{Lie}(\operatorname{O}(B))\) is precisely the Lie
+  subalgebra of \(\mathfrak{gl}(V)\) consisting of antisymmetric operators \(V
+  \to V\).
 \end{note}
 
 An interesting example of an invariant bilinear form is the so called