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: 43d2de9e0550c353493548639a975720bff9cd0f
Parent: 2f81acdfc052234279195fd9bb2946e70bc6addf
Author: Pablo <pablo-escobar@riseup.net>
Date: Fri, 13 Jan 2023 23:01:40 +0000

Fixed some typos

Diffstat

2 files changed, 19 insertions, 16 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/complete-reducibility.tex	18	10	8
Modified	sections/introduction.tex	17	9	8

diff --git a/sections/complete-reducibility.tex b/sections/complete-reducibility.tex
@@ -7,9 +7,9 @@ their representations are indeed useful, we are now faced with the Herculean
 task of trying to understand them. We have seen that representations can be
 used to derive geometric information about groups, but the question remains:
 how do we go about classifying the representations of a given Lie algebra? This
-is a question that have sparked an entire field of research, and we cannot hope
-to provide a comprehensive answer in the \pagedifference{start-47}{end-47}
-pages we have left. Nevertheless, we can work on particular cases.
+question has sparked an entire field of research, and we cannot hope to provide
+a comprehensive answer in the \pagedifference{start-47}{end-47} pages we have
+left. Nevertheless, we can work on particular cases.
 
 For instance, one can readily check that a \(K^n\)-module \(M\) -- here \(K^n\)
 denotes the \(n\)-dimensional Abelian Lie algebra -- is nothing more than a
@@ -820,7 +820,7 @@ establish\dots
       0 \rar & H^1(\mathfrak{g}, M) \rar & 0
     \end{tikzcd}
   \end{center}
-  then implies \(H^1(\mathfrak{g}, M) = 0\). Hence by induction in \(\dim V\)
+  then implies \(H^1(\mathfrak{g}, M) = 0\). Hence by induction in \(\dim M\)
   we find \(H^1(\mathfrak{g}, M) = 0\) for all finite-dimensional \(M\). We are
   done.
 \end{proof}
@@ -944,10 +944,12 @@ sequence
   \end{tikzcd}
 \end{center}
 
-This sequence always splits, which implies we can deduce information about
-\(\mathfrak{g}\)-modules by studying the modules of its ``semisimple
-part'' \(\mfrac{\mathfrak{g}}{\mathfrak{rad}(\mathfrak{g})}\) -- see
-Proposition~\ref{thm:quotients-by-rads}. In practice this translates to\dots
+This sequence always splits, which in light of
+Example~\ref{ex:all-simple-reps-are-tensor-prod} implies we can deduce
+information about \(\mathfrak{g}\)-modules by studying the modules of its
+``semisimple part'' \(\mfrac{\mathfrak{g}}{\mathfrak{rad}(\mathfrak{g})}\) --
+see Proposition~\ref{thm:quotients-by-rads}. In practice this translates
+to\dots
 
 \begin{proposition}[Lie]\label{thm:lie-thm-solvable-reps}
   Let \(\mathfrak{g}\) be a solvable Lie algebra. Every finite-dimensional

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -758,7 +758,7 @@ We should stress, however, that the representation theory of Lie algebras is
 only a small fragment of what is today known as ``representation theory'',
 which is in general concerned with a diverse spectrum of algebraic and
 combinatorial structures -- such as groups, quivers and associative algebras.
-An introductory exploration of some of this structures can be found in
+An introductory exploration of some of these structures can be found in
 \cite{etingof}.
 
 We begin by noting that any \(\mathcal{U}(\mathfrak{g})\)-module \(M\) may be
@@ -991,7 +991,10 @@ separate algebras. In particular, we may define\dots
 \begin{example}
   Given a Lie algebra \(\mathfrak{g}\), the adjoint \(\mathfrak{g}\)-module is
   a submodule of the restriction of the adjoint
-  \(\mathcal{U}(\mathfrak{g})\)-module to \(\mathfrak{g}\).
+  \(\mathcal{U}(\mathfrak{g})\)-module -- where we consider
+  \(\mathcal{U}(\mathfrak{g})\) a Lie algebra as in
+  Example~\ref{ex:inclusion-alg-in-lie-alg}, not as an associative algebra --
+  to \(\mathfrak{g}\).
 \end{example}
 
 Surprisingly, this functor has a right adjoint.
@@ -1085,9 +1088,7 @@ Example~\ref{ex:tensor-prod-separate-algs} thus provides a way to describe
 representations of \(\mathfrak{g} \oplus \mathfrak{h}\) in terms of the
 representations of \(\mathfrak{g}\) and \(\mathfrak{h}\). We will soon see that
 in many cases \emph{all} (simple) \(\mathfrak{g} \oplus \mathfrak{h}\)-modules
-can be constructed in such a manner.
-
-This concludes our initial remarks on \(\mathfrak{g}\)-modules. In the
-following chapters we will explore the fundamental problem of representation
-theory: that of classifying all representations of a given algebra up to
-isomorphism.
+can be constructed in such a manner. This concludes our initial remarks on
+\(\mathfrak{g}\)-modules. In the following chapters we will explore the
+fundamental problem of representation theory: that of classifying all
+representations of a given algebra up to isomorphism.