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

git clone: git://git.pablopie.xyz/lie-algebras-and-their-representations
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Commit: 07c9030a3f39d2d32691e9c68a9e37e70e0f7a61
Parent: b9b5ca882485dfe05be5e6319baf76a6a9606e04
Author: Pablo <pablo-escobar@riseup.net>
Date: Tue, 20 Dec 2022 19:41:45 +0000

Fixed some typos

Diffstat

3 files changed, 4 insertions, 4 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/complete-reducibility.tex	4	2	2
Modified	sections/introduction.tex	2	1	1
Modified	sections/sl2-sl3.tex	2	1	1

diff --git a/sections/complete-reducibility.tex b/sections/complete-reducibility.tex
@@ -211,7 +211,7 @@ clear things up.
 \end{proof}
 
 While are primarily interested in indecomposable representations -- which is
-usually a strickly larger class of representations than that of irreducible
+usually a strictly larger class of representations than that of irreducible
 representations -- it is important to note that irreducible representations are
 generally much easier to find. The relationship between irreducible
 representations is also well understood. This is because of the following
@@ -742,7 +742,7 @@ establish\dots
       \underbrace{C_V^{n - k} \pi(w)}_{= \; 0}
     = 0,
   \]
-  wich is a contradiction in light of the fact that neither \((-\lambda)^n\)
+  which is a contradiction in light of the fact that neither \((-\lambda)^n\)
   nor \(\pi(w)\) are nil. Hence \(V = W^\lambda\) and there must be some other
   eigenvalue \(\mu\) of \(C_V\!\restriction_W\). For any such \(\mu\) and any
   \(w \in W^\mu\),

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -657,7 +657,7 @@ we find\dots
 
 \begin{theorem}[Poincaré-Birkoff-Witt]
   Let \(\mathfrak{g}\) be a Lie algebra over \(K\) and \(\{X_i\}_i \subset
-  \mathfrak{g}\) be an orderer basis for \(\mathfrak{g}\) -- i.e. a basis
+  \mathfrak{g}\) be an ordered basis for \(\mathfrak{g}\) -- i.e. a basis
   indexed by an ordered set. Then \(\{X_{i_1} \cdot X_{i_2} \cdots X_{i_n} : n
   \ge 0, i_1 \le i_2 \le \cdots \le i_n\}\) is a basis for
   \(\mathcal{U}(\mathfrak{g})\).

diff --git a/sections/sl2-sl3.tex b/sections/sl2-sl3.tex
@@ -663,7 +663,7 @@ attentive reader may notice that \(B(E_{1 2}, E_{2 3}) = - \sfrac{1}{2}\), so
 that the angle -- with respect to the Killing form \(B\) -- between the root
 vectors \(E_{1 2}\) and \(E_{2 3}\) is precisely the same as the angle between
 the points representing their roots \(\alpha_1 - \alpha_2\) and \(\alpha_2 -
-\alpha_3\) in the Cartesion plane. Since \(\alpha_1 - \alpha_2\) and \(\alpha_2
+\alpha_3\) in the Cartesian plane. Since \(\alpha_1 - \alpha_2\) and \(\alpha_2
 - \alpha_3\) span \(\mathfrak{h}^*\), this implies the diagrams we've been
 drawing are given by an isometry \(\mathbb{Q} P \isoto \mathbb{Q}^2\), where
 \(\mathbb{Q} P\) is endowed with the bilinear form defined by \((\alpha_i -