Source code for my notes on representations of semisimple Lie algebras and Olivier Mathieu's classification of simple weight modules
Minor correction to the introduction
The radical and nilradical of a Lie algebra is only defined in the finite-dimensional case
1 file changed, 9 insertions, 9 deletions
| Status | File Name | N° Changes | Insertions | Deletions |
| Modified | sections/introduction.tex | 18 | 9 | 9 |
diff --git a/sections/introduction.tex b/sections/introduction.tex @@ -499,11 +499,11 @@ semisimple and reductive algebras by modding out by certain ideals, known as \emph{radicals}. \begin{definition}\index{Lie algebra!radical} - Let \(\mathfrak{g}\) be a Lie algebra. The sum \(\mathfrak{a} + - \mathfrak{b}\) of solvable ideals \(\mathfrak{a}, \mathfrak{b} \normal - \mathfrak{g}\) is again a solvable ideal. Hence the sum of all solvable - ideals of \(\mathfrak{g}\) is a maximal solvable ideal, known as \emph{the - radical \(\mathfrak{rad}(\mathfrak{g})\) of \(\mathfrak{g}\)}. + Let \(\mathfrak{g}\) be a finite-dimensioanl Lie algebra. The sum + \(\mathfrak{a} + \mathfrak{b}\) of solvable ideals \(\mathfrak{a}, + \mathfrak{b} \normal \mathfrak{g}\) is again a solvable ideal. Hence the sum + of all solvable ideals of \(\mathfrak{g}\) is a maximal solvable ideal, known + as \emph{the radical \(\mathfrak{rad}(\mathfrak{g})\) of \(\mathfrak{g}\)}. \[ \mathfrak{rad}(\mathfrak{g}) = \sum_{\substack{\mathfrak{a} \normal \mathfrak{g}\\\text{solvable}}} @@ -512,10 +512,10 @@ semisimple and reductive algebras by modding out by certain ideals, known as \end{definition} \begin{definition}\index{Lie algebra!nilradical} - Let \(\mathfrak{g}\) be a Lie algebra. The sum of nilpotent ideals is a - nilpotent ideal. Hence the sum of all nilpotent ideals of \(\mathfrak{g}\) is - a maximal nilpotent ideal, known as \emph{the nilradical - \(\mathfrak{nil}(\mathfrak{g})\) of \(\mathfrak{g}\)}. + Let \(\mathfrak{g}\) be a finite-dimensional Lie algebra. The sum of + nilpotent ideals is a nilpotent ideal. Hence the sum of all nilpotent ideals + of \(\mathfrak{g}\) is a maximal nilpotent ideal, known as \emph{the + nilradical \(\mathfrak{nil}(\mathfrak{g})\) of \(\mathfrak{g}\)}. \[ \mathfrak{nil}(\mathfrak{g}) = \sum_{\substack{\mathfrak{a} \normal \mathfrak{g}\\\text{nilpotent}}}