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

preamble.tex (6501B)

  1 \documentclass{book}
  2 \usepackage[total={6in, 9in}]{geometry}
  3 \usepackage{amsmath, amssymb, amsthm, stmaryrd, mathrsfs, gensymb, dsfont}
  4 \usepackage{mathtools, adjustbox}
  5 \usepackage{enumitem, xfrac, xcolor, cancel, multicol, tabularx, relsize}
  6 \usepackage[normalem]{ulem}
  7 \usepackage{hyperref, refcount}
  8 \usepackage{fancyhdr, titling}
  9 \usepackage[backend=biber, citetracker=true, citestyle=alphabetic, bibstyle=ieee-alphabetic, sorting=nty, sortcites=true]{biblatex}
 10 \usepackage{pgfplots, tikz, tikz-cd}
 11 \usepackage{graphicx, wrapfig}
 12 \usepackage[ordering=Kac]{dynkin-diagrams}
 13 \usepackage{rank-2-roots}
 14 \usepackage{imakeidx}
 15 \usepackage{tcolorbox}
 16 \usepackage{footnote}
 17 
 18 % Set the default colors
 19 \definecolor{mosgreen}{RGB}{29, 135, 17}
 20 \definecolor{thmbackground}{RGB}{249, 249, 249}
 21 \definecolor{thmrule}{RGB}{230, 230, 230}
 22 
 23 % Configure fonts
 24 \usepackage[T1]{fontenc}
 25 \usepackage[sc]{mathpazo}
 26 \renewcommand*\ttdefault{txtt}
 27 \usepackage[scr=euler,cal=boondox]{mathalfa}
 28 \hypersetup{
 29   colorlinks,
 30   citecolor=black,
 31   filecolor=black,
 32   linkcolor=black,
 33   urlcolor=mosgreen,
 34 }
 35 
 36 % Configure graphics
 37 \usetikzlibrary{calc, shadows.blur, shapes.geometric, patterns, arrows}
 38 \pgfplotsset{compat=1.16}
 39 
 40 % Initialize the imakeidx package
 41 \makeindex
 42 
 43 % Configure the style of Dynkin diagrams
 44 \tikzset{/Dynkin diagram,
 45          edge length=15mm,
 46          arrow shape/.style={-{To[length=7pt]}},
 47          mark=o,
 48          root-radius=.18cm,
 49          text style/.style={scale=1.2}}
 50 
 51 % Configure the enumerate environment to use bold roman numerals
 52 \setenumerate[0]{label={\color{mosgreen}\normalfont \bfseries(\roman*)}}
 53 
 54 % Macro for creating boxed theorem environments
 55 \newcounter{theorem}[chapter]
 56 \renewcommand{\thetheorem}{\thechapter.\arabic{theorem}}
 57 \newcommand{\newboxedtheorem}[2]
 58   {
 59     \newtheorem{inner#1}[theorem]{#2}
 60     \newenvironment{#1}[1][{}]
 61       {
 62         \savenotes
 63         \begin{tcolorbox}[colback=thmbackground, colframe=thmrule]
 64         \begin{inner#1}[##1]
 65       }
 66       {\end{inner#1}\end{tcolorbox}\spewnotes}
 67   }
 68 
 69 % Green theorem styles
 70 \newtheoremstyle{coloredtheorem}{}{}{\em}{}{\bf\color{mosgreen}}{.}{.5em}{}
 71 \newtheoremstyle{coloreddefinition}{}{}{}{}{\bf\color{mosgreen}}{.}{.5em}{}
 72 \newtheoremstyle{coloredremark}{}{}{}{}{\itshape\color{mosgreen}}{.}{.5em}{}
 73 
 74 % Useful theorem definitions
 75 \theoremstyle{coloredtheorem}
 76 \newboxedtheorem{theorem}{Theorem}
 77 \newboxedtheorem{lemma}{Lemma}
 78 \newboxedtheorem{corollary}{Corollary}
 79 \newboxedtheorem{proposition}{Proposition}
 80 \theoremstyle{coloreddefinition}
 81 \newboxedtheorem{definition}{Definition}
 82 \newtheorem{example}[theorem]{Example}
 83 \theoremstyle{coloredremark}
 84 \newtheorem*{note}{Remark}
 85 
 86 % Custom page style
 87 \fancypagestyle{custom}{%
 88     \fancyhead[EL]{%
 89     \raisebox{0.5em}{\thepage}%
 90     }
 91     \fancyhead[ER]{%
 92     \raisebox{0.5em}{\nouppercase{\leftmark}}%
 93     }
 94     \fancyhead[OL]{%
 95     \raisebox{0.5em}{\S\nouppercase{\rightmark}}%
 96     \vspace*{-0.5em}%
 97     }
 98     \fancyhead[OR]{%
 99     \raisebox{0.5em}{\thepage}%
100     }
101     \fancyfoot{}
102 }
103 \pagestyle{custom}
104 
105 % Dedication environment
106 \newenvironment{dedication}
107   {\clearpage            % We want a new page
108    \thispagestyle{empty} % No header and footer
109    \vspace*{\stretch{1}} % Some space at the top
110    \begin{center}        % Centralize text
111    \huge                 % Make the text larger
112    \itshape              % The text is in italics
113   }
114   {\par                  % End the paragraph
115    \end{center}
116    \vspace{\stretch{3}}  % Space at bottom is three times that at the top
117    \clearpage            % Finish off the page
118    \null
119    \thispagestyle{empty}
120    \clearpage            % Skip a page
121   }
122 
123 % Command to print licensing information
124 \newcommand{\printlicense}
125   {\clearpage
126    \thispagestyle{empty}
127    \vspace*{1cm} % Some space at the top
128    \begin{center}
129    This work is licensed under a \textbf{Creative Commons Attributions 4.0
130    International License}
131    \par
132    \vspace{1em}
133    \includegraphics[width=2cm]{images/by.eps}
134    \end{center}
135    \null\vfill
136    \begin{center}Last updated \today\end{center}
137    \clearpage
138   }
139 
140 % Command to define the subtitle in the cover of the book
141 \newcommand{\subtitle}[1]{\def\thesubtitle{#1}}
142 
143 % Use \blacksquare for \qed
144 \renewcommand{\qedsymbol}{\ensuremath{\blacksquare}}
145 
146 % Only use \smallsetminus
147 \renewcommand{\setminus}{\smallsetminus}
148 
149 % Get propper inequality symbols
150 \renewcommand{\leq}{\leqslant}
151 \renewcommand{\le}{\leqslant}
152 \renewcommand{\geq}{\geqslant}
153 \renewcommand{\ge}{\geqslant}
154 \renewcommand{\preceq}{\preccurlyeq}
155 \renewcommand{\succeq}{\succcurlyeq}
156 
157 % Add missing arrows
158 \newcommand{\longhookrightarrow}{\lhook\joinrel\longrightarrow}
159 \newcommand{\longhookleftarrow}{\longleftarrow\joinrel\rhook}
160 \DeclareRobustCommand\longtwoheadrightarrow
161   {\relbar\joinrel\twoheadrightarrow}
162 
163 % Fix the subset/supset symbols
164 \renewcommand{\subset}{\subseteq}
165 \renewcommand{\supset}{\supseteq}
166 
167 % A semantic alternative to \textbf
168 \newcommand{\strong}[1]{\textbf{#1}}
169 
170 % Display long arrows instead of short ones
171 \renewcommand{\to}{\longrightarrow}
172 \renewcommand{\mapsto}{\longmapsto}
173 
174 % Fix the goddamn \chi macro!
175 % For some reason the default LaTeX fonts place this character much lower than
176 % where people actually expect it to be. This new definition place it in with
177 % the appropriate spacing
178 \renewcommand{\chi}{\ensuremath \raisebox{\depth}{$\mathchar"11F$}}
179 
180 % Macro for Mathieu's coherent extension
181 \newcommand{\mExt}{\mathcal{E\!x\!t}}
182 
183 % Notation for twisted modules
184 \newcommand{\twisted}[2]{{}^{#2}\!{#1}}
185 
186 % Isomorphism arrow
187 \newcommand{\isoto}{\xlongrightarrow{\sim}}
188 
189 % Quotient object
190 \newcommand{\mfrac}[2]{\mathlarger{\sfrac{#1}{#2}}}
191 
192 % A normal subobject in a pointed cathegory
193 \newcommand{\normal}{\triangleleft}
194 
195 % Command for marking a node inside a matrix
196 \newcommand{\tm}[2]{%
197   \tikz[overlay,remember picture,baseline] \node [anchor=base] (#1) {$#2$};%
198 }
199 
200 % Command for drawing a vertical line between nodes in a matrix
201 \newcommand{\DrawVLine}[3][]{%
202   \begin{tikzpicture}[overlay,remember picture]
203     \draw[shorten <=0.3ex, #1] (#2.north) -- (#3.south);
204   \end{tikzpicture}
205 }
206 
207 % Command for drawing a horizontal line between nodes in a matrix
208 \newcommand{\DrawHLine}[3][]{%
209   \begin{tikzpicture}[overlay,remember picture]
210     \draw[shorten <=0.2em, #1] (#2.west) -- (#3.east);
211   \end{tikzpicture}
212 }
213 
214 % Command for printing the number of pages between two labels
215 \newcommand{\pagedifference}[2]{%
216   \number\numexpr\getpagerefnumber{#2}-\getpagerefnumber{#1}\relax}
217