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: 6bf0af16898b24e2a761309927bafc2b696aa806
Parent: e8ad33003404743ebafe04d4ced6eed85372117a
Author: Pablo <pablo-escobar@riseup.net>
Date: Wed, 31 Aug 2022 17:46:21 +0000

Ajustada a dedicatória, o título e o prefácio

Diffstat

3 files changed, 8 insertions, 47 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	preamble.tex	4	3	1
Modified	sections/preface.tex	44	3	41
Modified	tcc.tex	7	2	5

diff --git a/preamble.tex b/preamble.tex
@@ -82,10 +82,12 @@
   {\clearpage            % We want a new page
    \thispagestyle{empty} % No header and footer
    \vspace*{\stretch{1}} % Some space at the top
+   \begin{center}        % Centralize text
+   \huge                 % Make the text larger
    \itshape              % The text is in italics
-   \raggedleft           % Flush to the right margin
   }
   {\par                 % End the paragraph
+   \end{center}
    \vspace{\stretch{3}} % Space at bottom is three times that at the top
    \clearpage           % Finish off the page
   }

diff --git a/sections/preface.tex b/sections/preface.tex
@@ -1,34 +1,4 @@
-\chapter*{About This Notes}
-
-\begin{note}
-  Under construction!
-\end{note}
-
-These notes mostly amount to an amalgamation of thoughts and ideas I came
-across when studding the representation theory of groups. The primary focus of
-this notes is the beautiful interaction between algebra and geometry that occurs
-in representation theory. At first glance representation theory may seem like
-just another branch of abstract algebra. Historically, however, algebraic
-proofs in the representation theory of groups have been preceded by geometric
-proofs -- sometimes by several decades. This is something Georgie Williamson
-discusses at length in the excellent
-\citetitle{geometric-representation-williason}
-\cite{geometric-representation-williason}, but perhaps the idea is better
-synthesized in the eloquent words of Élie Cartan:
-
-\begin{displayquote}
-  \dots the difficulty, dare I not say the impossibility, of finding a proof
-  which does not leave the strict infinitesimal domain shows the necessity of
-  not sacrificing either point of view \dots
-\end{displayquote}
-
-The last quote is something Cartan wrote to Herman Weyl in 1925, after Weyl
-published his proof of complete reducibility of representations of complex
-semisimple Lie algebras -- those being ``the strict infinitesimal domain''.
-His proof relied heavily on Weyl's previous work on smooth representations of
-compact Lie groups, and a purely algebraic proof would only surface after about
-a decade. This is a particular example of the common phenomena described by
-Williamson.
+\chapter*{About This Text}
 
 Throughout this notes we'll follow the following guiding principles:
 
@@ -36,14 +6,6 @@ Throughout this notes we'll follow the following guiding principles:
   \item Lengthy proves are favored as opposed to collections of smaller lemmas.
     This is a deliberate effort to emphasize the relevant results.
 
-  \item Geometric proofs, as opposed to purely algebraic proofs, are generally
-    preferred. This is again a deliberate effort to emphasize the connections
-    between the geometric and the algebraic. We should clarify that when we say
-    \emph{geometric} we mean it in a very general sense -- basically anything
-    vaguely motivated by some notion of \emph{space}. That is to say, when we
-    say \emph{geometry} we don't necessarily mean \emph{differential geometry}
-    or \emph{algebraic geometry}.
-
   \item We prefer, whenever possible, to outsource proofs. This is because I
     don't fancy reinventing the wheel: I'll write down proofs in here
     \emph{only} when I fell like I have something to add to the proofs provided
@@ -52,6 +14,6 @@ Throughout this notes we'll follow the following guiding principles:
     statement of the theorem.
 \end{enumerate}
 
-We'll assume basic knowledge of abstract algebra, group theory and differential
-geometry. Additional topics will be covered when needed.
+We'll assume basic knowledge of abstract algebra and module theory. Additional
+topics will be covered when needed.

diff --git a/tcc.tex b/tcc.tex
@@ -1,8 +1,7 @@
 \input{./preamble.tex}
 \addbibresource{references.bib}
 
-\title{Representation Theory}
-\subtitle{The Algebraic \& the Geometric}
+\title{Lie Algebras \& their Representations}
 \author{Thiago Brevidelli Garcia}
 
 \begin{document}
@@ -10,9 +9,7 @@
 \input{cover}
 
 \begin{dedication}
-  This notes are dedicated to my dear friend Lucas Schiezari, who somehow
-  convinced me to apply for a bachelor's degree in pure mathematics. May he
-  rest in peace.
+  To Violeta
 \end{dedication}
 
 \pagenumbering{roman}