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