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, 11 insertions, 1 deletion

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -65,7 +65,17 @@
   T_1 G\). In particular, \(\mathfrak{g}\) is finite-dimensional.
 \end{example}
 
-% TODOO: Point out this construction "works" for algebraic groups too!
+\begin{example}
+  Let \(G\) be an algebraic \(K\)-group and \(K[G]\) denote the ring of regular
+  functions \(G \to K\). We call a derivation \(D : K[G] \to K[G]\) left
+  invariant if \(D(g \cdot f) = g \cdot D f\) for all \(g \in G\) and \(f \in
+  K[G]\) -- where the action of \(G\) in \(K[G]\) given by \((g \cdot f)(h) =
+  f(g^{-1} h)\). The commutator of left invariant derivations is invariant too,
+  so the space \(\operatorname{Lie}(G)\) of invariant derivations in \(K[G]\)
+  has the structure of a Lie algebra over \(K\) with brackets given by the
+  commutator of derivations. Again, \(\operatorname{Lie}(G) \cong T_1 G\) is
+  finite-dimensional.
+\end{example}
 
 \begin{example}
   The Lie algebra \(\operatorname{Lie}(\operatorname{GL}_n(K))\) is canonically