- Commit
- 293ed6c5742c63ef2e72f14441af442b865419fd
- Parent
- 7121a171332d9b0b4e03b770b39786ebbfc9c2ee
- Author
- Pablo <pablo-escobar@riseup.net>
- Date
Cleaned TODO items
Source code for my notes on representations of semisimple Lie algebras and Olivier Mathieu's classification of simple weight modules
Cleaned TODO items
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.