- Commit
- 854e67bcff0ee44a2d07eba9b3068fa65c2ccf5d
- Parent
- 9c254f28439268d83e9e55efc47a440391717f32
- Author
- Pablo <pablo-escobar@riseup.net>
- Date
Removed a stupid epigraph
Source code for my notes on representations of semisimple Lie algebras and Olivier Mathieu's classification of simple weight modules
Removed a stupid epigraph
1 file changed, 4 insertions, 8 deletions
Status | File Name | N° Changes | Insertions | Deletions |
Modified | sections/sl2-sl3.tex | 12 | 4 | 8 |
diff --git a/sections/sl2-sl3.tex b/sections/sl2-sl3.tex @@ -1,8 +1,5 @@ \chapter{Representations of \(\mathfrak{sl}_2(K)\) and \(\mathfrak{sl}_3(K)\)} -% TODO: Remove this? -\epigraph{Nobody has ever bet enough on a winning horse.}{Some gambler} - % TODOOOO: Write an intetroduction! \section{Representations of \(\mathfrak{sl}_2(K)\)}\label{sec:sl2} @@ -38,11 +35,10 @@ of \(V\) -- where \(\lambda\) ranges over the eigenvalues of \(h\) and \(V_\lambda\) is the corresponding eigenspace. At this point, this is nothing short of a gamble: why look at the eigenvalues of \(h\)? -The short answer is that, as we shall see, this will pay off -- which -conveniently justifies the epigraph of this chapter. For now we will postpone -the discussion about the real reason of why we chose \(h\). Let \(\lambda\) be -any eigenvalue of \(h\). Notice \(V_\lambda\) is in general not a -subrepresentation of \(V\). Indeed, if \(v \in V_\lambda\) then +The short answer is that, as we shall see, this will pay off. For now we will +postpone the discussion about the real reason of why we chose \(h\). Let +\(\lambda\) be any eigenvalue of \(h\). Notice \(V_\lambda\) is in general not +a subrepresentation of \(V\). Indeed, if \(v \in V_\lambda\) then \begin{align*} h e v & = 2e v + e h v = (\lambda + 2) e v \\ h f v & = - 2f v + f h v = (\lambda - 2) f v