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

Commit
293ed6c5742c63ef2e72f14441af442b865419fd
Parent
7121a171332d9b0b4e03b770b39786ebbfc9c2ee
Author
Pablo <pablo-escobar@riseup.net>
Date

Cleaned TODO items

Diffstat

2 files changed, 6 insertions, 8 deletions

Status File Name N° Changes Insertions Deletions
Modified sections/complete-reducibility.tex 7 1 6
Modified sections/sl2-sl3.tex 7 5 2
diff --git a/sections/complete-reducibility.tex b/sections/complete-reducibility.tex
@@ -1,10 +1,6 @@
 \chapter{Semisimplicity \& Complete Reducibility}
 
-% TODO: Remove this?
-\epigraph{Nobody has ever bet enough on a winning horse.}{Some gambler}
-
-% TODO: Update the 40 pages thing when we're done TODO: Have we seen the fact
-% representations are useful?
+% TODO: Update the 40 pages thing when we're done 
 Having hopefully established in the previous chapter that Lie algebras and
 their representations are indeed useful, we are now faced with the Herculean
 task of trying to understand them. We have seen that representations are a
@@ -626,7 +622,6 @@ a representation}.
   by \(\lambda_{i j}, \mu_{i j} \in K\) the coefficients of \(X_j\) and \(X^j\)
   in \([X, X_i]\) and \([X, X^i]\), respectively.
 
-  % TODO: Comment on the invariance of the Killing form beforehand
   The invariance of \(B_V\) implies
   \[
     \lambda_{i k}
diff --git a/sections/sl2-sl3.tex b/sections/sl2-sl3.tex
@@ -1,5 +1,10 @@
 \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}
 
 The primary goal of this section is proving\dots
@@ -1056,8 +1061,6 @@ The fundamental difference between these two cases is thus the fact that \(\dim
 question then is: why did we choose \(\mathfrak{h}\) with \(\dim \mathfrak{h} >
 1\) for \(\mathfrak{sl}_3(K)\)?
 
-% TODO: Add a note on how irreducible representations of Abelian algebras are
-% all one dimensional to the previous chapter
 The rational behind fixing an Abelian subalgebra is a simple one: we have seen
 in the previous chapter that representations of Abelian
 algebras are generally much simpler to understand than the general case.