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: 12047085f6e751cae156e4fb690b3f7ebeace90c
Parent: 7c14f2cb28b7341de7e556e6f840eb76710196e9
Author: Pablo <pablo-escobar@riseup.net>
Date: Sun, 27 Nov 2022 11:59:27 +0000

Removed duplicate words

Diffstat

3 files changed, 4 insertions, 4 deletions

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

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -740,7 +740,7 @@ over the ring of \(G\)-invariant differential operators -- i.e.
 \(G\).
 
 Proposition~\ref{thm:geometric-realization-of-uni-env} is in fact only the
-beginning of a profound connection between the theory of \(D\)-modules and and
+beginning of a profound connection between the theory of \(D\)-modules and
 \emph{representation theory}, the latter of which we now explore in the
 following section.

diff --git a/sections/semisimple-algebras.tex b/sections/semisimple-algebras.tex
@@ -926,8 +926,8 @@ are really interested in is\dots
 
 We should point out that proposition~\ref{thm:verma-is-finite-dim} fails for
 non-dominant \(\lambda \in P\). While \(\lambda\) is always a maximal weight of
-\(M(\lambda)\), one can show show that if \(\lambda \in P\) is not dominant
-then \(N(\lambda) = 0\) and \(M(\lambda)\) is irreducible. For instance, if
+\(M(\lambda)\), one can show that if \(\lambda \in P\) is not dominant then
+\(N(\lambda) = 0\) and \(M(\lambda)\) is irreducible. For instance, if
 \(\mathfrak{g} = \mathfrak{sl}_2(K)\) and \(\lambda : h \mapsto -2\) then the
 action of \(\mathfrak{g}\) on \(M(\lambda)\) is given by
 \begin{center}

diff --git a/sections/sl2-sl3.tex b/sections/sl2-sl3.tex
@@ -821,7 +821,7 @@ eigenvalue \(\lambda([E_{1 2}, E_{2 1}]) - 2k\) of the action of \(h\) on
 \(\bigoplus_{k \in \mathbb{N}} V_{\lambda + k (\alpha_1 - \alpha_2)}\) is
 \(V_{\lambda + k (\alpha_1 - \alpha_2)}\), the weights of \(V\) appearing the
 string \(\lambda, \lambda + (\alpha_1 - \alpha_2), \ldots, \lambda + k
-(\alpha_1 - \alpha_2), \ldots\) must be symmetric with respect to the the line
+(\alpha_1 - \alpha_2), \ldots\) must be symmetric with respect to the line
 \(B(\alpha_1 - \alpha_2, \alpha) =  0\). The picture is thus
 \begin{center}
   \begin{tikzpicture}