- Commit
- cd37967073dd160fe8c40a202553dfed5f7c8c11
- Author
- Pablo <pablo-escobar@riseup.net>
- Date
Commit inicial
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
Diffstat
8 files changed, 2449 insertions, 0 deletions
| Status | File Name | N° Changes | Insertions | Deletions |
| Added | .gitignore | 26 | 26 | 0 |
| Added | Makefile | 23 | 23 | 0 |
| Added | cover.tex | 79 | 79 | 0 |
| Added | preamble.tex | 184 | 184 | 0 |
| Added | references.bib | 171 | 171 | 0 |
| Added | sections/lie-algebras.tex | 1877 | 1877 | 0 |
| Added | sections/preface.tex | 57 | 57 | 0 |
| Added | tcc.tex | 32 | 32 | 0 |
diff --git a/.gitignore b/.gitignore @@ -0,0 +1,26 @@ +notes.epub +notes.pdf +*.aux +*.bak +*.bbl +*.blg +*.log +*.nav +*.out +*.xml +*.snm +*.toc +*.fdb_latexmk +*.fls +*.4ht +*.4ct +*.4tc +*.dvi +*.idv +*.lg +*.tmp +*.xdv +*.xref +*.pdf_original +images/*.pdf +archive.zip
diff --git a/Makefile b/Makefile @@ -0,0 +1,23 @@ +.PHONY: all compile auto-compile clean archive.zip + +all: compile + +# Generates a ZIP archive of the source code +archive.zip: + find ./ -name '*.tex' -or -name '*.bib' | xargs zip $@ + find images -type f | xargs zip -j $@ + +# Compile the document +compile: + latexmk tcc.tex + +# Compile the document whenever a (relevant) file is changed +auto-compile: + find ./ -name '*.tex' -or -name '*.bib' -or -name '*.tikz' \ + | entr make || exit 0 + +# Removed LuaLaTeX and ASpell artifacts +clean: + texclear tcc.tex + find . -name '*.bak' -delete +
diff --git a/cover.tex b/cover.tex @@ -0,0 +1,79 @@ +% Taken from https://latexdraw.com/tikz-cover-pages-gallery/ + +\definecolor{brickred}{rgb}{0.65, 0.16, 0.16} + +% Configure the PDF title and author +\ifdefined\thesubtitle +\hypersetup{ + pdftitle={\thetitle -- \thesubtitle}, + pdfauthor={{\theauthor}} +} +\else +\hypersetup{ + pdftitle={\thetitle}, + pdfauthor={\theauthor} +} +\fi + +% The cover itself +\thispagestyle{empty} +\begin{tikzpicture}[remember picture,overlay] + %%%%%%%%%%%%%%%%%%%% Background %%%%%%%%%%%%%%%%%%%%%%%% + \fill[brickred] (current page.south west) rectangle (current page.north east); + + + \foreach \i in {2.5,...,22} + { + \node[rounded corners,brickred!60,draw,regular polygon,regular polygon sides=6, minimum size=\i cm,ultra thick] at ($(current page.west)+(2.5,-5)$) {} ; + } + + %%%%%%%%%%%%%%%%%%%% Background Polygon %%%%%%%%%%%%%%%%%%%% + \foreach \i in {0.5,...,22} + { + \node[rounded corners,brickred!60,draw,regular polygon,regular polygon sides=6, minimum size=\i cm,ultra thick] at ($(current page.north west)+(2.5,0)$) {} ; + } + + \foreach \i in {0.5,...,22} + { + \node[rounded corners,brickred!90,draw,regular polygon,regular polygon sides=6, minimum size=\i cm,ultra thick] at ($(current page.north east)+(0,-9.5)$) {} ; + } + + + \foreach \i in {21,...,6} + { + \node[brickred!85,rounded corners,draw,regular polygon,regular polygon sides=6, minimum size=\i cm,ultra thick] at ($(current page.south east)+(-0.2,-0.45)$) {} ; + } + + + %%%%%%%%%%%%%%%%%%%% Title of the Report %%%%%%%%%%%%%%%%%%%% + \node[left,brickred!5,minimum width=0.625*\paperwidth,minimum height=3cm, rounded corners] at ($(current page.north east)+(0,-9.5)$) + { + \fontsize{22}{27} \selectfont \bfseries + \expandafter\MakeUppercase\expandafter{\thetitle} + }; + + %%%%%%%%%%%%%%%%%%%% Subtitle %%%%%%%%%%%%%%%%%%%% + \ifdefined\thesubtitle + \node[left,brickred!10,minimum width=0.625*\paperwidth,minimum height=2cm, rounded corners] at ($(current page.north east)+(0,-11)$) + { + {\huge \textsl{\thesubtitle}} + }; + \fi + + %%%%%%%%%%%%%%%%%%%% Author Name %%%%%%%%%%%%%%%%%%%% + \node[left,brickred!5,minimum width=0.625*\paperwidth,minimum height=2cm, rounded corners] at ($(current page.north east)+(0,-13)$) + { + {\Large \textsc{\theauthor}} + }; + + %%%%%%%%%%%%%%%%%%%% Year %%%%%%%%%%%%%%%%%%%% + \node[rounded corners,fill=brickred!70,text=brickred!5,regular polygon,regular polygon sides=6, minimum size=2.5 cm,inner sep=0,ultra thick] + at ($(current page.west)+(2.5,-5)$) {\LARGE \bfseries \number \year}; +\end{tikzpicture} + +% Back of the cover +\clearpage +\null +\thispagestyle{empty} +\addtocounter{page}{-2} +\clearpage
diff --git a/preamble.tex b/preamble.tex @@ -0,0 +1,184 @@ +\documentclass{book} +\usepackage[total={6in, 9in}]{geometry} +\usepackage{amsmath, amssymb, amsthm, stmaryrd, mathrsfs, gensymb, dsfont} +\usepackage{mathtools, adjustbox} +\usepackage[scr=esstix,cal=boondox]{mathalfa} +\usepackage{enumitem, xfrac, xcolor, cancel, multicol, tabularx, relsize} +\usepackage{fancyvrb, hyperref, csquotes} +\usepackage{xalgebra, xtopology, xgeometry, functional} +\usepackage{ytableau} +\usepackage[backend=biber]{biblatex} +\usepackage{pgfplots, tikz, tikz-cd} +\usepackage{graphicx, wrapfig} +\usepackage{xparse} +\usepackage[normalem]{ulem} +\usepackage{epigraph} +\usepackage[ordering=Kac]{dynkin-diagrams} +\usepackage{rank-2-roots} +\usepackage{fancyhdr} +\usepackage{titling} + +% Configure how link look +\hypersetup{ + colorlinks, + citecolor=black, + filecolor=black, + linkcolor=black +} + +% Configure graphics +\usetikzlibrary{calc, shadows.blur, shapes.geometric, patterns, arrows} +\pgfplotsset{compat=1.16} + +% Configure the style of Dynkin diagrams +\tikzset{/Dynkin diagram, + edge length=15mm, + arrow shape/.style={-{To[length=7pt]}}, + mark=o, + root-radius=.18cm, + text style/.style={scale=1.2}} + +% Configure the enumerate environment to use bold roman numerals +\setenumerate[0]{label={\normalfont \bfseries(\roman*)}} + +% Useful theorem definitions +\newtheorem{theorem}{Theorem}[section] +\newtheorem*{theorem*}{Theorem} +\newtheorem{lemma}{Lemma}[section] +\newtheorem*{lemma*}{Lemma} +\newtheorem{corollary}{Corollary}[section] +\newtheorem*{corollary*}{Corollary} +\newtheorem{proposition}{Proposition}[section] +\newtheorem*{proposition*}{Proposition} +\theoremstyle{remark} +\newtheorem*{note}{Remark} +\theoremstyle{definition} +\newtheorem{definition}{Definition}[section] +\newtheorem*{definition*}{Definition} +\newtheorem{example}{Example}[section] +\newtheorem*{example*}{Example} + +% Custom page style +\fancypagestyle{custom}{% + \fancyhead[EL]{% + \raisebox{0.5em}{\thepage}% + } + \fancyhead[ER]{% + \raisebox{0.5em}{\nouppercase{\leftmark}}% + } + \fancyhead[OL]{% + \raisebox{0.5em}{\S\nouppercase{\rightmark}}% + \vspace*{-0.5em}% + } + \fancyhead[OR]{% + \raisebox{0.5em}{\thepage}% + } + \fancyfoot{} +} +\pagestyle{custom} + +% Dedication environment +\newenvironment{dedication} + {\clearpage % We want a new page + \thispagestyle{empty} % No header and footer + \vspace*{\stretch{1}} % Some space at the top + \itshape % The text is in italics + \raggedleft % Flush to the right margin + } + {\par % End the paragraph + \vspace{\stretch{3}} % Space at bottom is three times that at the top + \clearpage % Finish off the page + } + +% Command to define the subtitle in the cover of the book +\newcommand{\subtitle}[1]{\def\thesubtitle{#1}} + +% Double struck letters +% The following is kept besides \AA being already defined because I don't care +% abaut Å +\renewcommand{\AA}{\mathbb{A}} % Double struck A +\newcommand{\BB}{\mathbb{B}} % Double struck B +\newcommand{\CC}{\mathbb{C}} % Double struck C +\newcommand{\DD}{\mathbb{D}} % Double struck D +\newcommand{\EE}{\mathbb{E}} % Double struck E +\newcommand{\FF}{\mathbb{F}} % Double struck F +\newcommand{\GG}{\mathbb{G}} % Double struck G +\newcommand{\HH}{\mathbb{H}} % Double struck H +\newcommand{\II}{\mathbb{I}} % Double struck I +\newcommand{\JJ}{\mathbb{J}} % Double struck J +\newcommand{\KK}{\mathbb{K}} % Double struck K +\newcommand{\LL}{\mathbb{L}} % Double struck L +\newcommand{\MM}{\mathbb{M}} % Double struck M +\newcommand{\NN}{\mathbb{N}} % Double struck N +\newcommand{\OO}{\mathbb{O}} % Double struck O +\newcommand{\PP}{\mathbb{P}} % Double struck P +\newcommand{\QQ}{\mathbb{Q}} % Double struck Q +\newcommand{\RR}{\mathbb{R}} % Double struck R +% The following is ommited because LaTeX keeps complaining about \SS not being +% allowed in mathmode +% \renewcommand{\SS}{\mathbb{S}} % Double struck S +\newcommand{\TT}{\mathbb{T}} % Double struck T +\newcommand{\UU}{\mathbb{U}} % Double struck U +\newcommand{\VV}{\mathbb{V}} % Double struck V +\newcommand{\WW}{\mathbb{W}} % Double struck W +\newcommand{\XX}{\mathbb{X}} % Double struck X +\newcommand{\YY}{\mathbb{Y}} % Double struck Y +\newcommand{\ZZ}{\mathbb{Z}} % Double struck Z +\newcommand{\hh}{\mathds{h}} % Double struck h +\newcommand{\kk}{\mathds{k}} % Double struck k + +% Use \blacksquare for \qed +\renewcommand{\qedsymbol}{\ensuremath{\blacksquare}} + +% Only use \smallsetminus +\renewcommand{\setminus}{\smallsetminus} + +% Make the \mid symbol taller (this is useful for Group Theory) +\renewcommand{\mid}{\,\mathlarger{\mathlarger{\mathchar"326A}}\,} +\renewcommand{\nmid}{\,\mathlarger{\mathlarger{\mathchar"352D}}\,} + +% Get propper inequality symbols +\renewcommand{\leq}{\leqslant} +\renewcommand{\le}{\leqslant} +\renewcommand{\geq}{\geqslant} +\renewcommand{\ge}{\geqslant} +\renewcommand{\preceq}{\preccurlyeq} +\renewcommand{\succeq}{\succcurlyeq} + +% Add missing arrows +\newcommand{\longhookrightarrow}{\lhook\joinrel\longrightarrow} +\newcommand{\longhookleftarrow}{\longleftarrow\joinrel\rhook} +\DeclareRobustCommand\longtwoheadrightarrow + {\relbar\joinrel\twoheadrightarrow} + +% Fix the subset symbol +\renewcommand{\subset}{\subseteq} + +% Redefine \bar +\renewcommand{\bar}{\overline} + +% A semantic alternative to \textbf +\newcommand{\strong}[1]{\textbf{#1}} + +% Set cardinal +\newcommand{\card}[1]{\left|\nobreak#1\nobreak\right|} + +% Minus sign with a dot over it. Usefull for typsetting saturated subtraction +\def\dotminus{\mathbin{\ooalign{\hss\raise1ex\hbox{.}\hss\cr + \mathsurround=0pt$-$}}} + +% Empty macro (it's usefull for typesetting Young diagrams) +\newcommand{\void}{ } + +% Display long arrows instead of short ones +\renewcommand{\to}{\longrightarrow} +\renewcommand{\mapsto}{\longmapsto} +\renewcommand{\To}{\Longrightarrow} + +% Fix the goddamn \chi and \wp macros! +% For some reason the default LaTeX fonts place this characters much lower than +% where people actually expect them to be. This new definitions place them in +% with the appropriate spacing +\renewcommand{\chi}{\ensuremath \raisebox{\depth}{$\mathchar"11F$}} +\renewcommand{\wp}{\ensuremath \raisebox{\depth}{$\mathchar"17D$}} +
diff --git a/references.bib b/references.bib @@ -0,0 +1,171 @@ +@book{serganova, + title = {A Journey Through Representation Theory: From Finite Groups to Quivers via Algebras}, + author = {Caroline Gruson, Vera Serganova}, + publisher = {}, + year = {2018}, + series = {}, + edition = {}, + volume = {}, +} + +@book{alexandrino, + title = {Notas de Aula de MAT5771}, + author = {Marcos Alexandrino}, + year = {2019}, + url = {https://www.ime.usp.br/~malex/arquivos/lista2019/GeoRiemanniana-Main-novo2019.pdf}, +} + +@book{fulton-harris, + title = {Representation theory: A first course}, + author = {William Fulton, Joe Harris}, + publisher = {Springer}, + year = {1991}, + series = {Graduate Texts in Mathematics / Readings in Mathematics}, + edition = {Corrected}, + volume = {}, +} + +@book{san-martin-algebras, + title = {Álgebras de Lie}, + author = {Luiz Antonio Barrera San Martin}, + publisher = {Unicamp}, + year = {1999}, + series = {}, + edition = {1}, + volume = {}, +} + +@book{san-martin-grupos, + title = {Grupos de Lie}, + author = {Luiz A. B. San Martin}, + publisher = {Unicamp}, + year = {2016}, + series = {}, + edition = {}, + volume = {}, +} + +@book{procesi, + title = {Lie Groups: An Approach through Invariants and Representations}, + author = {Claudio Procesi}, + publisher = {Springer}, + year = {2006}, + series = {Universitext}, + edition = {1}, + volume = {}, +} + +@book{etingof, + title = {Introduction to Representation Theory}, + author = {Pavel Etingof}, + publisher = {American Mathematical Society}, + year = {2011}, + series = {Student Mathematical Library}, +} + +@book{fourier-on-number-fields, + title = {Fourier Analysis on Number Fields}, + author = {Dinakar Ramakrishnan, Robert J. Valenza}, + publisher = {Springer}, + year = {1998}, + series = {Graduate Texts in Mathematics v. 186}, + edition = {1}, + volume = {}, +} + +@book{lie-theorems, + title = {LECTURE 12: LIE'S FUNDAMENTAL THEOREMS}, + author = {Zuoqin Wang}, + year = {2013}, + url = {http://staff.ustc.edu.cn/~wangzuoq/Courses/13F-Lie/Lie.html} +} + +@book{gorodski, + title = {Lecture Notes on Compact Lie Groups and Their Representation -- Prelimimary version}, + author = {Claudio Gorodski}, + year = {2021}, + url = {https://www.ime.usp.br/~gorodski/teaching/mat6001/16.11.pdf} +} +@book{intro-to-lie-groups, + title = {Introduction to Lie groups}, + author = {Joseph Hundley}, + publisher = {Southern Illinois University Carbondale}, + year = {2009}, + series = {Miscellaneous (presentations, translations, interviews, etc)}, + edition = {}, + volume = {Paper 42}, + url = {https://opensiuc.lib.siu.edu/math_misc/42/} +} + +@book{fegan, + title = {Introduction To Compact Lie Groups}, + author = {Howard D Fegan}, + publisher = {World Scientific}, + isbn = {9810236867}, + year = {1991}, + series = {Series in Pure Mathematics}, +} + +@book{kirillov, + title = {Introduction to Lie groups and Lie algebras}, + author = {Alexander Kirillov}, + publisher = {Cambridge University Press}, + year = {2008}, + series = {}, + url = {https://www.math.stonybrook.edu/~kirillov/liegroups}, +} + +@book{lie-groups-serganova-student, + title = {261A Lie Groups}, + author = {Qiaochu Yuan}, + year = {2013}, + series = {}, + url = {https://math.berkeley.edu/~qchu/Notes/261A.pdf}, +} + +@misc{geometric-representation-williason, + title = {Representation theory and geometry }, + author = {Geordie Williamson}, + year = {2018}, + publisher = {2018 Internatition Congress of Mathematicians}, + series = {Plenary Lectures}, + url = {https://www.youtube.com/watch?v=-3q6C558yog}, +} + +@article{moore, + doi = {10.2140/pjm.1962.12.359}, + year = {1962}, + month = mar, + publisher = {Mathematical Sciences Publishers}, + volume = {12}, + number = {1}, + pages = {359--365}, + author = {Calvin Moore}, + title = {On the Frobenius reciprocity theorem for locally compact groups}, + journal = {Pacific Journal of Mathematics} +} + +@book{cx-lie, + title = {Notes on complex Lie groups}, + author = {Dietmar A. Salamon}, + year = {2019}, + url = {https://people.math.ethz.ch/~salamon/PREPRINTS/cx-lie.pdf} +} + +@book{humphreys, + title = {Introduction to Lie algebras and representation theory}, + author = {J.E. Humphreys}, + publisher = {Springer}, + year = {1973}, + series = {Graduate Texts in Mathematics}, + edition = {1}, +} + +@article{unitary-group-strong-topology, + doi = {10.48550/ARXIV.1309.5891}, + author = {Schottenloher, Martin}, + title = {The Unitary Group In Its Strong Topology}, + publisher = {arXiv}, + year = {2013}, +} +
diff --git a/sections/lie-algebras.tex b/sections/lie-algebras.tex @@ -0,0 +1,1877 @@ +\chapter{Semisimple Lie Algebras \& their Representations}\label{ch:lie-algebras} + +\epigraph{Nobody has ever bet enough on a winning horse.}{\textit{Some +gambler}} + +We've just established that \(\Rep(G) \cong \Rep(\mathfrak{g}_\CC)\) for each +simply connected \(G\), but how do we go about classifying the representations +of \(\mathfrak{g}_\CC\)? We can very quickly see that many of the aspects of +the representation theory of compact groups that were essential for the +solution our classification problem are no longer valid in this context. For +instance if we take +\[ + \mathfrak g = + \begin{pmatrix} + \CC & \CC \\ + 0 & 0 + \end{pmatrix} + \subset \gl_2(\CC) +\] +and consider its natural representation \(V = \CC^2\) we can see that \(W = +\langle (1, 0) \rangle\) is a subrepresentation with no \(\mathfrak +g\)-invariant complement. The issue we now face is, of course, the fact that +complete reducibility fails for some complex Lie algebras. In other words, +understanding the irreducible representations of an algebra isn't enough, +because there may be representations which cannot be expressed as the direct +sum of irreducible ones. + +% TODO: Update the "two chapter!" thing after those chapters are done? +This is not going to be an easy ride\dots \ In fact, even after some further +restrictions we will soon impose, this classification +problem will keep us busy for the next \emph{two} chapters! The primary goal +of this chapter is to highlight this restrictions and study some particular +examples in the hopes of getting some insight into the general case. We should +point that the following discussion is \emph{immensely} inspired by the third +part of \citetitle{fulton-harris} \cite{fulton-harris}: Fulton \& Harris are +the real authors of this chapter, I am merely a commentator on their work. + +The restriction we'll make is, as you might have guessed, that we'll primarily +focus on \emph{semisimple Lie algebras} -- those for whom \emph{complete +reducibility}, also known as \emph{semisimplicity}, holds. +This is a bit of an admission of defeat from my end, as we won't ever get to +classify the smooth representations of \emph{all} simply connected Lie groups, +but keep in mind that the problem of classifying the finite-dimensional +representations of an arbitrary finite-dimensional Lie algebra is still an open +one. In other words, the semisimple Lie algebras are the only algebras whose +representation we can expect to understand in full generality. This goes to +show the importance of complete reducibility in representation theory. + +I guess we could simply define semisimple Lie algebras as the class of complex +Lie algebras whose representations are completely reducible, but this is about +as satisfying as saying ``the semisimple are the ones who won't cause us any +trouble''. Who are the semisimple Lie algebras? Why does complete reducibility +holds for them? + +\section{Semisimplicity \& Complete Reducibility} + +There are multiple equivalent ways to define what a semisimple Lie algebra is, +the most obvious of which we have already mentioned in the above. Perhaps the +most common definition is\dots + +\begin{definition}\label{thm:sesimple-algebra} + A Lie algebra \(\mathfrak g\) is called \emph{semisimple} if it has no + non-zero solvable ideals -- i.e. subalgebras \(\mathfrak h\) with + \([\mathfrak h, \mathfrak g] \subset \mathfrak h\) whose derived series + \[ + \mathfrak h + \supseteq [\mathfrak h, \mathfrak h] + \supseteq [[\mathfrak h, \mathfrak h], [\mathfrak h, \mathfrak h]] + \supseteq + [ + [[\mathfrak h, \mathfrak h], [\mathfrak h, \mathfrak h]], + [[\mathfrak h, \mathfrak h], [\mathfrak h, \mathfrak h]] + ] + \supseteq \cdots + \] + converges to \(0\) in finite time. +\end{definition} + +\begin{example} + The complex Lie algebras \(\sl_n(\CC)\) and \(\mathfrak{sp}_{2 n}(\CC)\) are + both semisimple -- see the section of \cite{kirillov} on invariant bilinear + forms and the semisimplicity of classical Lie algebras. +\end{example} + +A popular alternative to definition~\ref{thm:sesimple-algebra} is\dots + +\begin{definition}\label{def:semisimple-is-direct-sum} + A Lie algebra \(\mathfrak g\) is called semisimple if it is the direct sum of + simple Lie algebras -- i.e. Lie algebras \(\mathfrak s\) whose only ideals + are \(0\) and \(\mathfrak s\). +\end{definition} + +\begin{example} + Every Abelian Lie algebra \(\mathfrak{g}\) is semisimple. Indeed, if \(\{X_i + : i \in \Lambda\}\) is a basis of \(\mathfrak{g}\) then + \[ + \mathfrak{g} = \bigoplus_i \CC X_i + \] + as a vector space. Clearly, each \(\CC X_i\) is a subalgebra of + \(\mathfrak{g}\). But \(\dim \CC X_i = 1\), so that the only subspaces of + \(\CC X_i\) are \(0\) and \(\CC X_i\). In particular, the only ideals of + \(\CC X_i\) are \(0\) and \(\CC X_i\) -- i.e. \(\CC X_i\) is simple. + Moreover, the fact that \([X_i, X_j] = 0\) for all \(i\) and \(j\) implies + \(\mathfrak{g} \cong \bigoplus_i \CC X_i\) as a Lie algebra. +\end{example} + +\begin{example} + The algebra \(\gl_n(\CC) = \sl_n(\CC) \oplus \CC\) is semisimple. In fact, + the direct sum of any two semisimple Lie algebras is semisimple. +\end{example} + +I suppose this last definition explains the nomenclature, but what does any of +this have to do with complete reducibility? Well, the special thing about +semisimple complex Lie algebras is that they are \emph{compact algebras}. +Compact Lie algebras are, as you might have guessed, \emph{algebras that come +from compact groups}. In other words\dots + +\begin{theorem}\label{thm:compact-form} + If \(\mathfrak g\) is a complex semisimple Lie algebra, there exists a + (unique) semisimple real form of \(\mathfrak g\) whose simply connected form + is compact. +\end{theorem} + +The proof of theorem~\ref{thm:compact-form} is quite involved and we will only +provide a rough outline -- mostly based on the proof by \cite{fegan} -- but the +interesting thing about it is the fact that we have already classified the +smooth (complex) representations of compact Lie groups: because of +chapter~\ref{ch:compacts}, we already know every smooth representation of a +compact group is completely reducible. We can then use this knowledge to lift +complete reducibility to our semisimple algebra. For instance, given a complex +representation \(V\) of \(\sl_n(\CC)\) and a subrepresentation \(W \subset V\), +\begin{enumerate} + \item Since \(\mathfrak{su}_n \otimes \CC \cong \sl_n(\CC)\), \(V\) and \(W\) + correspond to complex representations of \(\mathfrak{su}_n\) + \item Since \(\SU_n\) is simply connected, \(V\) and \(W\) correspond to + smooth representations of \(\SU_n\) + \item Since \(\SU_n\) is compact, \(W\) admits a \(\SU_n\)-invariant + complement \(U \subset V\) + \item \(U\) is invariant under the action of \(\mathfrak{su}_n\), so\dots + \item \(U\) is invariant under the action of \(\sl_n(\CC)\) and + therefore\dots + \item \(U\) is a \(\sl_n(\CC)\)-invariant complement of \(W\) in \(V\) +\end{enumerate} + +If we assume theorem~\ref{thm:compact-form} and replace \(\sl_n(\CC)\) with +some arbitrary semisimple \(\mathfrak{g}\) in the previous paragraph we arrive +at a proof of\dots + +\begin{theorem} + Every representation of a semisimple Lie algebra is completely reducible. +\end{theorem} + +By the same token, most of other aspects of representation +theory of compact groups must hold in the context of semisimple algebras. For +instance, we have\dots + +\begin{lemma}[Schur] + Let \(V\) and \(W\) be two irreducible representations of a complex + semisimple Lie algebra \(\mathfrak{g}\) and \(T : V \to W\) be an + intertwining operator. Then either \(T = 0\) or \(T\) is an isomorphism. + Furthermore, if \(V = W\) then \(T\) is scalar multiple of the identity. +\end{lemma} + +\begin{corollary} + Every irreducible representation of an Abelian Lie group is 1-dimensional. +\end{corollary} + +Indeed, if we take theorem~\ref{thm:compact-form} at face +value we can easily see that given a semisimple Lie algebra \(\mathfrak g\), +\[ + \Rep(\mathfrak g) + \cong \Rep(\mathfrak{g}_\RR) + \cong \Rep(G), +\] +where \(\mathfrak{g}_\RR\) is some real form of \(\mathfrak g\) with compact +simply connected form \(G\). This is what's known as \emph{Weyl's unitarization +trick}, and historically it was the first proof of complete reducibility for +semisimple Lie algebras. + +Alternatively, one could prove the same statement in a purely algebraic manner +by showing the first Lie algebra cohomology group \(H^1(\mathfrak{g}, V) = +\Ext^1(\CC, V)\) vanishes for all \(V\), as do \cite{kirillov} and +\cite{lie-groups-serganova-student} in their proofs. More precisely, one can +show that there is a natural bijection between \(H^1(\mathfrak{g}, \Hom(V, +W))\) and isomorphism classes of the representations \(U\) of \(\mathfrak{g}\) +such that there is an exact sequence +\begin{center} + \begin{tikzcd} + 0 \arrow{r} & + V \arrow{r} & + U \arrow{r} & + W \arrow{r} & + 0 + \end{tikzcd} +\end{center} + +This implies every exact sequence of \(\mathfrak{g}\)-representations splits -- +which, if you recall theorem~\ref{thm:complete-reducibility-equiv}, is +equivalent to complete reducibility -- if, and only if \(H^1(\mathfrak{g}, +\Hom(V, W)) = 0\) for all \(V\) and \(W\). The algebraic approach has the +advantage of working for Lie algebras over arbitrary fields, but in keeping +with our principle of preferring geometric arguments over purely algebraic one +we'll instead focus in the unitarization trick. What follows is a sketch of its +proof, whose main ingredient is\dots + +\section{The Killing Form} + +\begin{definition} + Given a -- either real or complex -- Lie algebra, its Killing form is the + symmetric bilinear form + \[ + K(X, Y) = \Tr(\ad(X) \ad(Y)) + \] +\end{definition} + +The Killing form certainly deserves much more attention than what we can +afford at the present moment, but what's relevant to us is the fact that +theorem~\ref{thm:compact-form} can be deduced from an algebraic condition +satisfied by the Killing forms of complex semisimple algebras. Explicitly\dots + +\begin{theorem}\label{thm:killing-form-is-negative} + If \(\mathfrak g\) is semisimple then there exists a semisimple real Lie + algebra \(\mathfrak{g}_\RR\) whose complexification is precisely \(\mathfrak + g\) and whose Killing form is negative-definite. +\end{theorem} + +The proof of theorem~\ref{thm:killing-form-is-negative} is combinatorial in +nature and it can be found in chapter 26 of \cite{fulton-harris}. What we're +interested in at the moment is showing it implies +theorem~\ref{thm:compact-form}. We'll start out by showing\dots + +\begin{lemma} + If \(\mathfrak{g}_\RR\) is a real Lie algebra with negative-definite Killing + form and \(G\) is its simply connected form then \(\mfrac{G}{Z(G)}\) is + compact. +\end{lemma} + +\begin{proof} + Let \(G\) be the simply connected form of \(\mathfrak{g}_\RR\). Consider the + the adjoint action \(\Ad : G \to \Aut(\mathfrak{g}_\RR)\). + + We'll start by point out that given \(g \in G\), + \[ + \begin{split} + K(X, Y) + & = \Tr(\ad(X) \ad(Y)) \\ + & = \Tr(\Ad(g) (\ad(X) \ad(Y)) \Ad(g)^{-1}) \\ + & = \Tr((\Ad(g) \ad(X) \Ad(g)^{-1}) (\Ad(g) \ad(Y) \Ad(g)^{-1})) \\ + \text{(because \(\Ad(g)\) is a homomorphism)} + & = \Tr(\ad(\Ad(g) X) \ad(\Ad(g) Y)) \\ + & = K(\Ad(g) X, \Ad(g) Y)) + \end{split} + \] + + Now since \(K\) is negative-definite, \(\Ad(g)\) is an orthogonal operator. + Hence \(\Ad(G)\) is a closed subgroup of \(\operatorname{O}(n)\) -- where \(n + = \dim \mathfrak{g}_\RR\). Notice \(Z(G) = \ker \Ad\). Indeed, if \(\Ad(g) = + \Id\) by corollary~\ref{thm:lie-group-morphism-at-identity} + \(h \mapsto g h g^{-1}\) is the identity map -- i.e. \(g \in Z(G)\). It then + follows from the fact that \(\operatorname{O}(n)\) is compact that + \[ + \mfrac{G}{Z(G)} + = \mfrac{G}{\ker \Ad} + \cong \Ad(G) + \] + is compact. +\end{proof} + +We should point out that this last trick can also be used to prove that +\(\mathfrak{g}_\RR\) is the direct sum of simple algebras. Indeed, if +\(\mathfrak{g}_\RR\) is not simple then, by definition, it has a proper +subalgebra \(\mathfrak h\). We can then consider its orthogonal complement +\(\mathfrak{h}^\perp\) under the Killing form, so that \(\mathfrak{h}^\perp\) +is a subalgebra and \(\mathfrak{g}_\RR = \mathfrak{h} \oplus +\mathfrak{h}^\perp\). Now by induction on the dimension of \(\mathfrak{g}_\RR\) +we see that theorem~\ref{thm:killing-form-is-negative} implies the +characterization of definition~\ref{def:semisimple-is-direct-sum}. + +To conclude this dubious attempt at a proof, we refer to a theorem by Hermann +Weyl, whose proof is beyond the scope of this notes as it requires calculating +the Ricci curvature of \(G\) \footnote{The Ricci curvature is a tensor related +to any given connection in a manifold. In this proof we're interested in the +Ricci curvature of the Riemannian connection of \(\widetilde H\) under the +metric given by the pullback of the unique bi-invariant metric of \(H\) along +the covering map \(\widetilde H \to H\).} -- for a proof please refer to +theorem 3.2.15 of \cite{gorodski}. What's interesting about this theorem is it +implies\dots + +\begin{theorem}[Weyl] + If \(H\) is a compact connected Lie group with discrete center then its + universal cover \(\widetilde H\) is also compact. +\end{theorem} + +\begin{proof}[Proof of theorem~\ref{thm:compact-form}] + Let \(\mathfrak{g}_\RR\) be a semisimple real form of \(\mathfrak g\) with + negative-definite Killing form. Because of the previous lemma, we already + know \(\mfrac{G}{Z(G)}\) is compact and centerless. Hence by Weyl's theorem + it suffices to show \(Z(G) = \ker \Ad\) is discrete -- so that the universal + cover of \(\mfrac{G}{Z(G)}\) is \(G\). + + To do so, we consider its Lie algebra \(\mathfrak z = \ker \ad\) -- also + known as the center of \(\mathfrak{g}_\RR\). Notice \(\mathfrak z\) is an + ideal. In fact, \(\mathfrak z\) is a solvable ideal of \(\mathfrak{g}_\RR\) + -- indeed, \([\mathfrak z, \mathfrak z] = 0\). This implies \(\mathfrak z = + 0\) and therefore \(Z(G)\) is a 0-dimensional Lie group -- i.e. a discrete + group. We are done. +\end{proof} + +This results can be generalized to a certain extent by considering the exact +sequence +\begin{center} + \begin{tikzcd} + 0 \arrow{r} & + \Rad(\mathfrak g) \arrow{r} & + \mathfrak g \arrow{r} & + \mfrac{\mathfrak g}{\Rad(\mathfrak g)} \arrow{r} & + 0 + \end{tikzcd} +\end{center} +where \(\Rad(\mathfrak g)\) is the sum of all solvable ideals of \(\mathfrak +g\) -- i.e. a maximal solvable ideal -- for arbitrary complex \(\mathfrak g\). +This implies we can deduce information about the representations of \(\mathfrak +g\) by studying those of its semisimple part \(\mfrac{\mathfrak +g}{\Rad(\mathfrak g)}\). In practice though, this isn't quite satisfactory +because the exactness of this last sequence translates to the +underwhelming\dots + +\begin{theorem}\label{thm:semi-simple-part-decomposition} + Every irreducible representation of \(\mathfrak g\) is the tensor product of + an irreducible representation of its semisimple part \(\mfrac{\mathfrak + g}{\Rad(\mathfrak g)}\) and a one-dimensional representation of \(\mathfrak + g\). +\end{theorem} + +We say that this isn't satisfactory because +theorem~\ref{thm:semi-simple-part-decomposition} is a statement about +\emph{irreducible} representations of \(\mathfrak g\). This may sound a bit +unfair, as theorem~\ref{thm:semi-simple-part-decomposition} does lead to a +complete classification of a large class of representations of \(\mathfrak g\) +-- those that are the direct sum of irreducible representations -- but the +point is that these may not be all possible representations if \(\mathfrak g\) +is not semisimple. That said, we can finally get to the classification itself. +Without further ado, we'll start out by highlighting a concrete example of the +general paradigm we'll later adopt: that of \(\sl_2(\CC)\). + +\section{Representations of \(\sl_2(\CC)\)} + +The primary goal of this section is proving\dots + +\begin{theorem}\label{thm:sl2-exist-unique} + For each \(n > 0\), there exists precisely one irreducible representation + \(V\) of \(\sl_2(\CC)\) with \(\dim V = n\). +\end{theorem} + +It's important to note, however, that -- as promised -- we will end up with an +explicit construction of \(V\). The general approach we'll take is supposing +\(V\) is an irreducible representation of \(\sl_2(\CC)\) and then derive some +information about its structure. We begin our analysis by pointing out that +the elements +\begin{align*} + e & = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} & + f & = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix} & + h & = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} +\end{align*} +form a basis of \(\sl_2(\CC)\) and satisfy +\begin{align*} + [e, f] & = h & [h, f] & = -2 f & [h, e] = 2 e +\end{align*} + +This is interesting to us because it implies every subspace of \(V\) invariant +under the actions of \(e\), \(f\) and \(h\) has to be \(V\) itself. Next we +turn our attention to the action of \(h\) in \(V\), in particular, to the +eigenspace decomposition +\[ + V = \bigoplus_{\lambda} V_\lambda +\] +of \(V\) -- where \(\lambda\) ranges over the eigenvalues of \(h\) and +\(V_\lambda\) is the corresponding eigenspace. At this point, this is nothing +short of a gamble: why look at the eigenvalues of \(h\)? + +The short answer is that, as we shall see, this will pay off -- which +conveniently justifies the epigraph of this chapter. For now we will postpone +the discussion about the real reason of why we chose \(h\). Let \(\lambda\) be +any eigenvalue of \(h\). Notice \(V_\lambda\) is in general not a +subrepresentation of \(V\). Indeed, if \(v \in V_\lambda\) then +\begin{align*} + h e v & = 2e v + e h v = (\lambda + 2) e v \\ + h f v & = - 2f v + f h v = (\lambda - 2) f v +\end{align*} + +In other words, \(e\) sends an element of \(V_\lambda\) to an element of +\(V_{\lambda + 2}\), while \(f\) sends it to an element of \(V_{\lambda - 2}\). +Hence +\begin{center} + \begin{tikzcd} + \cdots \arrow[bend left=60]{r} + & V_{\lambda - 2} \arrow[bend left=60]{r}{e} \arrow[bend left=60]{l} + & V_{\lambda} \arrow[bend left=60]{r}{e} \arrow[bend left=60]{l}{f} + & V_{\lambda + 2} \arrow[bend left=60]{r} \arrow[bend left=60]{l}{f} + & \cdots \arrow[bend left=60]{l} + \end{tikzcd} +\end{center} +and \(\bigoplus_{n \in \ZZ} V_{\lambda + 2 n}\) is an \(\sl_2(\CC)\)-invariant +subspace. This implies +\[ + V = \bigoplus_{n \in \ZZ} V_{\lambda + 2 n}, +\] +so that the eigenvalues of \(h\) all have the form \(\lambda + 2 n\) for some +\(n\) -- since \(V_\mu = 0\) for all \(\mu \notin \lambda + 2 \ZZ\). + +Even more so, if \(a = \min \{ n \in \ZZ : V_{\lambda + 2 n} \ne 0 \}\) and +\(b = \max \{ n \in \ZZ : V_{\lambda + 2 n} \ne 0 \}\) we can see that +\[ + \bigoplus_{\substack{n \in \ZZ \\ a \le n \le b}} V_{\lambda + 2 n} +\] +is also an \(\sl_2(\CC)\)-invariant subspace, so that the eigenvalues of \(h\) +form an unbroken string +\[ + \ldots, \lambda - 4, \lambda - 2, \lambda, \lambda + 2, \lambda + 4, \ldots +\] +around \(\lambda\). + +Our main objective is to show \(V\) is determined by this string of +eigenvalues. To do so, we suppose without any loss in generality that +\(\lambda\) is the right-most eigenvalue of \(h\), fix some non-zero \(v \in +V_\lambda\) and consider the set \(\{v, f v, f^2, v, \ldots\}\). + +\begin{theorem}\label{thm:basis-of-irr-rep} + The set \(\{v, f v, f^2, \ldots\}\) is a basis for \(V\). +\end{theorem} + +\begin{proof} + First of all, notice \(f^k v\) lies in \(V_{\lambda - 2 k}\), so that \(\{v, + f v, f^2 v, \ldots\}\) is a set of linearly independent vectors. Hence it + suffices to show \(V = \langle v, f v, f^2 v, \ldots \rangle\), which in + light of the fact that \(V\) is irreducible is the same as showing \(\langle + v, f v, f^2 v, \ldots \rangle\) is invariant under the action of + \(\sl_2(\CC)\). + + The fact that \(h f^k v \in \langle v, f v, f^2 v, \ldots \rangle\) follows + immediately from our previous assertion that \(f^k v \in V_{\lambda - 2 k}\) + -- indeed, \(h f^k v = (\lambda - 2 k) f^k v\). Seeing \(e f^k v \in \langle + v, f v, f^2 v, \ldots \rangle\) is a bit more complex. Clearly, + \[ + \begin{split} + e f v + & = h v + f e v \\ + \text{(since \(\lambda\) is the right-most eigenvalue)} + & = h v + f 0 \\ + & = \lambda v + \end{split} + \] + + Next we compute + \[ + \begin{split} + e f^2 v + & = (h + fe) f v \\ + & = h f v + f (\lambda v) \\ + & = 2 (\lambda - 1) f v + \end{split} + \] + + The pattern is starting to become clear: \(e\) sends \(f^k v\) to a multiple + of \(f^{k - 1} v\). Explicitly, it's not hard to check by induction that + \[ + e f^k v = k (\lambda + 1 - k) f^{k - 1} v + \] +\end{proof} + +\begin{note} + For this last formula to work we fix the convention that \(f^{-1} v = 0\) -- + which is to say \(e v = 0\). +\end{note} + +Theorem~\ref{thm:basis-of-irr-rep} may seem unrelated to our problem at first, +but it's significance lies in the fact that we have just provided a complete +description of the action of \(\sl_2(\CC)\) in \(V\). In other words\dots + +\begin{corollary} + \(V\) is completely determined by the right-most eigenvalue \(\lambda\) of + \(h\). +\end{corollary} + +\begin{proof} + If \(W\) is an irreducible representation of \(\sl_2(\CC)\) whose + right-most eigenvalue of \(h\) is \(\lambda\) and \(w \in W_\lambda\) is + non-zero, consider the linear isomorphism + \begin{align*} + T : V & \to W \\ + f^k v & \mapsto f^k w + \end{align*} + + We claim \(T\) is an intertwining operator. Indeed, the explicit calculations + of \(e f^k v\) and \(h f^k v\) from the previous proof imply + \begin{align*} + T e & = e T & T f & = f T & T h & = h T + \end{align*} +\end{proof} + +Other important consequences of theorem~\ref{thm:basis-of-irr-rep} are\dots + +\begin{corollary} + Every \(h\) eigenspace is one-dimensional. +\end{corollary} + +\begin{proof} + It suffices to note \(\{v, f v, f^2 v, \ldots \}\) is a basis for \(V\) + consisting of eigenvalues of \(h\) and whose only element in \(V_{\lambda - 2 + k}\) is \(f^k v\). +\end{proof} + +\begin{corollary} + The eigenvalues of \(h\) in \(V\) form a symmetric, unbroken string of + integers separated by intervals of length \(2\) whose right-most value is + \(\dim V - 1\). +\end{corollary} + +\begin{proof} + If \(f^m\) is the lowest power of \(f\) that annihilates \(v\), it follows + from the formula for \(e f^k v\) obtained in the proof of + theorem~\ref{thm:basis-of-irr-rep} that + \[ + 0 = e 0 = e f^m v = m (\lambda + 1 - m) f^{m - 1} v + \] + + This implies \(\lambda + 1 - m = 0\) -- i.e. \(\lambda = m - 1 \in \ZZ\). Now + since \(\{v, f v, f^2 v, \ldots, f^{m - 1} v\}\) is a basis for \(V\), \(m = + \dim V\). Hence if \(n = \lambda = \dim V - 1\) then the eigenvalues of \(h\) + are + \[ + \ldots, n - 6, n - 4, n - 2, n + \] + + To see that this string is symmetric around \(0\), simply note that the + left-most eigenvalue of \(h\) is precisely \(n - 2 (m - 1) = -n\). +\end{proof} + +We now know every irreducible representation \(V\) of \(\sl_2(\CC)\) has the +form +\begin{center} + \begin{tikzcd} + \cdots \arrow[bend left=60]{r} + & V_{n - 6} \arrow[bend left=60]{r}{e} \arrow[bend left=60]{l} + & V_{n - 4} \arrow[bend left=60]{r}{e} \arrow[bend left=60]{l}{f} + & V_{n - 2} \arrow[bend left=60]{r}{e} \arrow[bend left=60]{l}{f} + & V_n \arrow[bend left=60]{l}{f} + \end{tikzcd} +\end{center} +where \(V_{n - 2 k}\) is the one-dimensional eigenspace of \(h\) associated to +\(n - 2 k\) and \(n = \dim V - 1\). Even more so, we explicitly know +\[ + V = \bigoplus_{k = 0}^n \CC f^k v +\] +and +\begin{equation}\label{eq:irr-rep-of-sl2} + \begin{aligned} + f^k v & \overset{e}{\mapsto} k(n + 1 - k) f^{k - 1} v + & f^k v & \overset{f}{\mapsto} f^{k + 1} v + & f^k v & \overset{h}{\mapsto} (n - 2 k) f^k v + \end{aligned} +\end{equation} + +To conclude our analysis all it's left is to show that for each \(n\) such +\(V\) does indeed exist and is irreducible. In other words\dots + +\begin{theorem}\label{thm:irr-rep-of-sl2-exists} + For each \(n \ge 0\) there exists a (unique) irreducible representation of + \(\sl_2(\CC)\) whose left-most eigenvalue of \(h\) is \(n\). +\end{theorem} + +\begin{proof} + The fact the representation \(V\) from the previous discussion exists is + clear from the commutator relations of \(\sl_2(\CC)\) -- just look at \(f^k + v\) as abstract symbols and impose the action given by + (\ref{eq:irr-rep-of-sl2}). Alternatively, one can readily check that if + \(\CC^2\) is the natural representation of \(\sl_2(\CC)\), then \(V = \Sym^n + \CC^2\) satisfies the relations of (\ref{eq:irr-rep-of-sl2}). To see that + \(V\) is irreducible let \(W\) be a non-zero subrepresentation and take some + non-zero \(w \in W\). Suppose \(w = \alpha_0 v + \alpha_1 f v + \cdots + + \alpha_n f^n v\) and let \(k\) be the lowest index such that \(\alpha_k \ne + 0\), so that + \[ + w = \alpha_k f^k v + \cdots + \alpha_n f^n v + \] + + Now given that \(f^m = f^{n + 1}\) annihilates \(v\), + \[ + f w = \alpha_k f^{k + 1} v + \cdots + \alpha_{n - 1} f^n v + \] + + Proceeding inductively we arrive at \(f^{n - k} w = \alpha_k f^n v\), so + that \(f^n v \in W\). Hence \(e^i f^n v = \prod_{k = 1}^i k(n + 1 - k) f^{n - + i} v \in W\) for all \(i = 1, 2, \ldots, n\). Since \(k \ne 0 \ne n + 1 - k\) + for all \(k\) in this range, we can see that \(f^k v \in W\) for all \(k = 0, + 1, \ldots, n\). In other words, \(W = V\). We are done. +\end{proof} + +A perhaps more elegant way of proving theorem~\ref{thm:irr-rep-of-sl2-exists} +is to work our way backwards to the corresponding representation \(V\) of +\(\SU_2\) and then show that its character is irreducible. Computing the +action of \(\mathfrak{su}_2\) in \(V\) is fairly easy: if we consider the +standard basis +\begin{align*} + I & = \begin{pmatrix} i & 0 \\ 0 & - i \end{pmatrix} & + J & = \begin{pmatrix} 0 & 1 \\ - 1 & 0 \end{pmatrix} & + K & = \begin{pmatrix} 0 & - i \\ - i & 0 \end{pmatrix} +\end{align*} +of \(\mathfrak{su}_2\) one can readily check +\begin{align*} + f^k v + & \overset{I}{\mapsto} + i (n - 2 k) f^k v \\ + f^k v + & \overset{J}{\mapsto} + k (n + 1 - k) f^{k - 1} v - f^{k + 1} v \\ + f^k v + & \overset{K}{\mapsto} + - i k (n + 1 - k) f^{k - 1} v - i f^{k + 1} v +\end{align*} + +If we denote by \(\rho_*(X)\) the matrix corresponding to the action of \(X \in +\mathfrak{su}_2\) in the basis \(\{v, f v, \ldots, f^n v\}\), and given that +\(\exp : \mathfrak{su}_2 \to \SU_2\) is surjective, to arrive at the action of +\(\SU_2\) all we have to do is compute \(\exp(\rho_*(X))\) for arbitrary \(X\). +For instance, if \(n = 1\) we can very quickly check that \(V\) corresponds to +the natural representation of \(\SU_2\), which can be shown to be irreducible. + +In general, however, computing \(\exp(\rho_*(X))\) is \emph{quite hard}. +Nevertheless, the exceptional isomorphism \(\mathbb{S}^3 \cong \SU_2\) allows +us to compute its trace: if \(a, b, c \in \RR\) are such that \(a^2 + b^2 + c^2 += 1\), then the eigenvalues of \(\rho_*(a I + b J + c K)\) are \(\lambda i\), +where \(\lambda\) ranges over the eigenvalues of \(h\) in \(V\). Hence the +eigenvalues of \(\exp(\rho_*(t X))\) are \(e^{\lambda i t}\) for \(X = a I + +b J + c K\). Now if \(p = a i + b j + c k \in \mathbb{S}^3\) then \(\cos t + p +\sin t = \exp(t X)\), so that +\[ + \chi_V(\cos t + p \sin t) + = \sum_\lambda e^{\lambda i t} + = \sum_\lambda \cos(\lambda t) +\] + +A simple calculation then shows +\[ + \norm{\chi_V}^2 + = \frac{1}{\mu(\mathbb{S}^3)} \int_{\mathbb{S}^3} \abs{\chi_V(q)}^2 \; \dd q + = 1, +\] +which establishes that \(V\) is irreducible. + +Our initial gamble of studying the eigenvalues of \(h\) may have seemed +arbitrary at first, but it payed off: we've \emph{completely} described +\emph{all} irreducible representations of \(\sl_2(\CC)\). It is not yet clear, +however, if any of this can be adapted to a general setting. In the following +section we shall double down on our gamble by trying to reproduce some of the +results of this section for \(\sl_3(\CC)\), hoping this will \emph{somehow} +lead us to a general solution. In the process of doing so we'll learn a bit +more why \(h\) was a sure bet and the race was fixed all along. + +\section{Representations of \(\sl_3(\CC)\)}\label{sec:sl3-reps} + +The study of representations of \(\sl_2(\CC)\) reminds me of the difference the +derivative of a function \(\RR \to \RR\) and that of a smooth map between +manifolds: it's a simpler case of something greater, but in some sense it's too +simple of a case, and the intuition we acquire from it can be a bit misleading +in regards to the general one. For instance I distinctly remember my Calculus I +teacher telling the class ``the derivative of the composition of two functions +is not the composition of their derivatives'' -- which is, of course, the +\emph{correct} formulation of the chain rule in the context of smooth +manifolds. + +The same applies to \(\sl_2(\CC)\). It's a simple and beautiful example, but +unfortunately the general picture -- representations of arbitrary semisimple +algebras -- lacks its simplicity, and, of course, much of this complexity is +hidden in the case of \(\sl_2(\CC)\). The general purpose of this section is +to investigate to which extent the framework used in the previous section to +classify the representations of \(\sl_2(\CC)\) can be generalized to other +semisimple Lie algebras, and the algebra \(\sl_3(\CC)\) stands as a natural +candidate for potential generalizations: \(3 = 2 + 1\) after all. + +Our approach is very straightforward: we'll fix some irreducible +representation \(V\) of \(\sl_3(\CC)\) and proceed step by step, at each point +asking ourselves how we could possibly adapt the framework we laid out for +\(\sl_2(\CC)\). The first obvious question is one we have already asked +ourselves: why \(h\)? More specifically, why did we choose to study its +eigenvalues and is there an analogue of \(h\) in \(\sl_3(\CC)\)? + +The answer to the former question is one we'll discuss at length in the +next chapter, but for now we note that perhaps the most fundamental +property of \(h\) is that \emph{there exists an eigenvector \(v\) of +\(h\) that is annihilated by \(e\)} -- that being the generator of the +right-most eigenspace of \(h\). This was instrumental to our explicit +description of the irreducible representations of \(\sl_2(\CC)\) culminating in +theorem~\ref{thm:irr-rep-of-sl2-exists}. + +Our fist task is to find some analogue of \(h\) in \(\sl_3(\CC)\), but it's +still unclear what exactly we are looking for. We could say we're looking for +an element of \(V\) that is annihilated by some analogue of \(e\), but the +meaning of \emph{some analogue of \(e\)} is again unclear. In fact, as we shall +see, no such analogue exists and neither does such element. Instead, the +actual way to proceed is to consider the subalgebra +\[ + \mathfrak h + = \left\{ + X \in + \begin{pmatrix} \CC & 0 & 0 \\ 0 & \CC & 0 \\ 0 & 0 & \CC \end{pmatrix} + : \Tr(X) = 0 + \right\} +\] + +The choice of \(\mathfrak{h}\) may seem like an odd choice at the moment, but +the point is we'll later show that there exists some \(v \in V\) that is +simultaneously an eigenvector of each \(H \in \mathfrak{h}\) and annihilated by +half of the remaining elements of \(\sl_3(\CC)\). This is exactly analogous to +the situation we found in \(\sl_2(\CC)\): \(h\) corresponds to the subalgebra +\(\mathfrak{h}\), and the eigenvalues of \(h\) in turn correspond to linear +functions \(\lambda : \mathfrak{h} \to \CC\) such that \(H v = \lambda(H) \cdot +v\) for each \(H \in \mathfrak{h}\) and some non-zero \(v \in V\). We call such +functionals \(\lambda\) \emph{eigenvalues of \(\mathfrak{h}\)}, and we say +\emph{\(v\) is an eigenvector of \(\mathfrak h\)}. + +Once again, we'll pay special attention to the eigenvalue decomposition +\begin{equation}\label{eq:weight-module} + V = \bigoplus_\lambda V_\lambda +\end{equation} +where \(\lambda\) ranges over all eigenvalues of \(\mathfrak{h}\) and +\(V_\lambda = \{ v \in V : H v = \lambda(H) \cdot v, \forall H \in \mathfrak{h} +\}\). We should note that the fact that (\ref{eq:weight-module}) is not at all +obvious. This is because in general \(V_\lambda\) is not the eigenspace +associated with an eigenvalue of any particular operator \(H \in +\mathfrak{h}\), but instead the eigenspace of the action of the entire algebra +\(\mathfrak{h}\). Fortunately for us, (\ref{eq:weight-module}) always holds, +but we will postpone its proof to the next chapter. + +Next we turn our attention to the remaining elements of \(\sl_3(\CC)\). In our +analysis of \(\sl_2(\CC)\) we saw that the eigenvalues of \(h\) differed from +one another by multiples of \(2\). A possible way to interpret this is to say +\emph{the eigenvalues of \(h\) differ from one another by integral linear +combinations of the eigenvalues of the adjoint action of \(h\)}. In English, +the eigenvalues of of the adjoint actions of \(h\) are \(\pm 2\) since +\begin{align*} + [h, f] & = -2 f & + [h, e] & = 2 e +\end{align*} +and the eigenvalues of the action of \(h\) in an irreducible +\(\sl_2(\CC)\)-representation differ from one another by multiples of \(\pm +2\). + +In the case of \(\sl_3(\CC)\), a simple calculation shows that if \([H, X]\) is +scalar multiple of \(X\) for all \(H \in \mathfrak{h}\) then all but one entry +of \(X\) are zero. Hence the eigenvectors of the adjoint action of +\(\mathfrak{h}\) are \(E_{i j}\) and its eigenvalues are \(\alpha_i - +\alpha_j\), where +\[ + \alpha_i + \begin{pmatrix} + a_1 & 0 & 0 \\ + 0 & a_2 & 0 \\ + 0 & 0 & a_3 + \end{pmatrix} + = a_i +\] + +Visually we may draw + +\begin{figure}[h] + \centering + \begin{tikzpicture}[scale=2.5] + \begin{rootSystem}{A} + \filldraw[black] \weight{0}{0} circle (.5pt); + \node[black, above right] at \weight{0}{0} {\small$0$}; + \wt[black]{-1}{2} + \wt[black]{-2}{1} + \wt[black]{1}{1} + \wt[black]{-1}{-1} + \wt[black]{2}{-1} + \wt[black]{1}{-2} + \node[above] at \weight{-1}{2} {$\alpha_2 - \alpha_3$}; + \node[left] at \weight{-2}{1} {$\alpha_2 - \alpha_1$}; + \node[right] at \weight{1}{1} {$\alpha_1 - \alpha_3$}; + \node[left] at \weight{-1}{-1} {$\alpha_3 - \alpha_1$}; + \node[right] at \weight{2}{-1} {$\alpha_1 - \alpha_2$}; + \node[below] at \weight{1}{-2} {$\alpha_3 - \alpha_1$}; + \node[black, above] at \weight{1}{0} {$\alpha_1$}; + \node[black, above] at \weight{-1}{1} {$\alpha_2$}; + \node[black, above] at \weight{0}{-1} {$\alpha_3$}; + \filldraw[black] \weight{1}{0} circle (.5pt); + \filldraw[black] \weight{-1}{1} circle (.5pt); + \filldraw[black] \weight{0}{-1} circle (.5pt); + \end{rootSystem} + \end{tikzpicture} +\end{figure} + +If we denote the eigenspace of the adjoint action of \(\mathfrak{h}\) in +\(\sl_3(\CC)\) associated to \(\alpha\) by \(\sl_3(\CC)_\alpha\) and fix some +\(X \in \sl_3(\CC)_\alpha\), \(H \in \mathfrak{h}\) and \(v \in V_\lambda\) +then +\[ + \begin{split} + H (X v) + & = X (H v) + [H, X] v \\ + & = X (\lambda(H) \cdot v) + (\alpha(H) \cdot X) v \\ + & = (\alpha + \lambda)(H) \cdot X v + \end{split} +\] +so that \(X\) carries \(v\) to \(V_{\alpha + \lambda}\). In other words, +\(\sl_3(\CC)_\alpha\) \emph{acts on \(V\) by translating vectors between +eigenspaces}. + +For instance \(\sl_3(\CC)_{\alpha_1 - \alpha_3}\) will act on the adjoint +representation of \(\sl_3(\CC)\) via +\begin{figure}[h] + \centering + \begin{tikzpicture}[scale=2.5] + \begin{rootSystem}{A} + \wt[black]{0}{0} + \wt[black]{-1}{2} + \wt[black]{-2}{1} + \wt[black]{1}{1} + \wt[black]{-1}{-1} + \wt[black]{2}{-1} + \wt[black]{1}{-2} + \draw[-latex, black] \weight{-1.9}{1.1} -- \weight{-1.1}{1.9}; + \draw[-latex, black] \weight{-.9}{-.9} -- \weight{-.1}{-.1}; + \draw[-latex, black] \weight{0.1}{0.1} -- \weight{.9}{.9}; + \draw[-latex, black] \weight{1.1}{-1.9} -- \weight{1.9}{-1.1}; + \end{rootSystem} + \end{tikzpicture} +\end{figure} + +This is again entirely analogous to the situation we observed in +\(\sl_2(\CC)\). In fact, we may once more conclude\dots + +\begin{theorem}\label{thm:sl3-weights-congruent-mod-root} + The eigenvalues of the action of \(\mathfrak{h}\) in an irreducible + \(\sl_3(\CC)\)-representation \(V\) differ from one another by integral + linear combinations of the eigenvalues \(\alpha_i - \alpha_j\) of + adjoint action of \(\mathfrak{h}\) in \(\sl_3(\CC)\). +\end{theorem} + +\begin{proof} + This proof goes exactly as that of the analogous statement for + \(\sl_2(\CC)\): it suffices to note that if we fix some eigenvalue + \(\lambda\) of \(\mathfrak{h}\) and let \(i\) and \(j\) vary then + \[ + \bigoplus_{i j} V_{\lambda + \alpha_i - \alpha_j} + \] + is an invariant subspace of \(V\). +\end{proof} + +To avoid confusion we better introduce some notation to differentiate between +eigenvalues of the action of \(\mathfrak{h}\) in \(V\) and eigenvalues of the +adjoint action of \(\mathfrak{h}\). + +\begin{definition} + Given a representation \(V\) of \(\sl_3(\CC)\), we'll call the non-zero + eigenvalues of the action of \(\mathfrak{h}\) in \(V\) \emph{weights of + \(V\)}. As you might have guessed, we'll correspondingly refer to + eigenvectors and eigenspaces of a given weight by \emph{weight vectors} and + \emph{weight spaces}. +\end{definition} + +It's clear from our previous discussion that the weights of the adjoint +representation of \(\sl_3(\CC)\) deserve some special attention. + +\begin{definition} + The weights of the adjoint representation of \(\sl_3(\CC)\) are called + \emph{roots of \(\sl_3(\CC)\)}. Once again, the expressions \emph{root + vector} and \emph{root space} are self-explanatory. +\end{definition} + +Theorem~\ref{thm:sl3-weights-congruent-mod-root} can thus be restated as\dots + +\begin{corollary} + The weights of an irreducible representation \(V\) of \(\sl_3(\CC)\) are all + congruent module the lattice \(Q\) generated by the roots \(\alpha_i - + \alpha_j\) of \(\sl_3(\CC)\). +\end{corollary} + +\begin{definition} + The lattice \(Q = \ZZ \langle \alpha_i - \alpha_j : i, j = 1, 2, 3 \rangle\) + is called \emph{the root lattice of \(\sl_3(\CC)\)}. +\end{definition} + +To proceed we once more refer to the previously established framework: next we +saw that the eigenvalues of \(h\) formed an unbroken string of integers +symmetric around \(0\). To prove this we analyzed the right-most eigenvalue of +\(h\) and its eigenvector, providing an explicit description of the +irreducible representation of \(\sl_2(\CC)\) in terms of this vector. We may +reproduce these steps in the context of \(\sl_3(\CC)\) by fixing a direction in +the place an considering the weight lying the furthest in that direction. + +In practice this means we'll choose a linear functional \(f : \mathfrak{h}^* +\to \RR\) and pick the weight that maximizes \(f\). To avoid any ambiguity we +should choose the direction of a line irrational with respect to the root +lattice \(Q\). For instance if we choose the direction of \(\alpha_1 - +\alpha_3\) and let \(f\) be the projection \(Q \to \RR \langle \alpha_1 - +\alpha_3 \rangle \cong \RR\) then \(\alpha_1 - 2 \alpha_2 + \alpha_3 \in Q\) +lies in \(\ker f\), so that if a weight \(\lambda\) maximizes \(f\) then the +translation of \(\lambda\) by any multiple of \(\alpha_1 - 2 \alpha_2 + +\alpha_3\) must also do so. In others words, if the direction we choose is +parallel to a vector lying in \(Q\) then there may be multiple choices the +``weight lying the furthest'' along this direction. + +Let's say we fix the direction +\begin{center} + \begin{tikzpicture}[scale=2.5] + \begin{rootSystem}{A} + \wt[black]{0}{0} + \wt[black]{-1}{2} + \wt[black]{-2}{1} + \wt[black]{1}{1} + \wt[black]{-1}{-1} + \wt[black]{2}{-1} + \wt[black]{1}{-2} + \draw[-latex, black, thick] \weight{-1.5}{-.5} -- \weight{1.5}{.5}; + \end{rootSystem} + \end{tikzpicture} +\end{center} +and let \(\lambda\) be the weight lying the furthest in this direction. + +\begin{definition} + We say that a root \(\alpha\) is positive if \(f(\alpha) > 0\) -- i.e. if it + lies to the right of the direction we chose. Otherwise we say \(\alpha\) is + negative. Notice that \(f(\alpha) \ne 0\) since by definition \(\alpha \ne + 0\) and \(f\) is irrational with respect to the lattice \(Q\). +\end{definition} + +The first observation we make is that all others weights of \(V\) must lie in a +sort of \(\frac{1}{3}\)-plane with corners at \(\lambda\), as shown in +\begin{center} + \begin{tikzpicture} + \AutoSizeWeightLatticefalse + \begin{rootSystem}{A} + \weightLattice{3} + \fill[gray!50,opacity=.2] (hex cs:x=5,y=-7) -- (hex cs:x=1,y=1) -- + (hex cs:x=-7,y=5) arc (150:270:{7*\weightLength}); + \draw[black, thick] (hex cs:x=5,y=-7) -- (hex cs:x=1,y=1) -- + (hex cs:x=-7,y=5); + \filldraw[black] (hex cs:x=1,y=1) circle (1pt); + \node[above right=-2pt] at (hex cs:x=1,y=1) {\small\(\lambda\)}; + \end{rootSystem} + \end{tikzpicture} +\end{center} + +Indeed, if this is not the case then, by definition, \(\lambda\) is not the +furthest weight along the line we chose. Given our previous assertion that the +root spaces of \(\sl_3(\CC)\) act on the weight spaces of \(V\) via +translation, this implies that \(E_{1 2}\), \(E_{1 3}\) and \(E_{2 3}\) all +annihilate \(V_\lambda\), or otherwise one of \(V_{\lambda + \alpha_1 - +\alpha_2}\), \(V_{\lambda + \alpha_1 - \alpha_3}\) and \(V_{\lambda + \alpha_2 - +\alpha_3}\) would be non-zero -- which contradicts the hypothesis that +\(\lambda\) lies the furthest along the direction we chose. In other words\dots + +\begin{theorem} + There is a weight vector \(v \in V\) that is killed by all positive root + spaces of \(\sl_3(\CC)\). +\end{theorem} + +\begin{proof} + It suffices to note that the positive roots of \(\sl_3(\CC)\) are precisely + \(\alpha_1 - \alpha_2\), \(\alpha_1 - \alpha_3\) and \(\alpha_2 - \alpha_3\). +\end{proof} + +We call \(\lambda\) \emph{the highest weight of \(V\)}, and we call any \(v \in +V_\lambda\) \emph{a highest weight vector}. Going back to the case of +\(\sl_2(\CC)\), we then constructed an explicit basis of our irreducible +representations in terms of a highest weight vector, which allowed us to +provide an explicit description of the action of \(\sl_2(\CC)\) in terms of +its standard basis and finally we concluded that the eigenvalues of \(h\) must +be symmetrical around \(0\). An analogous procedure could be implemented for +\(\sl_3(\CC)\) -- and indeed that's what we'll do later down the line -- but +instead we would like to focus on the problem of finding the weights of \(V\) +for the moment. + +We'll start out by trying to understand the weights in the boundary of +\(\frac{1}{3}\)-plane previously drawn. Since the root spaces act by +translation, the action of \(E_{2 1}\) in \(V_\lambda\) will span a subspace +\[ + W = \bigoplus_k V_{\lambda + k (\alpha_2 - \alpha_1)}, +\] +and by the same token \(W\) must be invariant under the action of \(E_{1 2}\). + +To draw a familiar picture +\begin{center} + \begin{tikzpicture} + \begin{rootSystem}{A} + \node at \weight{3}{1} (a) {}; + \node at \weight{1}{2} (b) {}; + \node at \weight{-1}{3} (c) {}; + \node at \weight{-3}{4} (d) {}; + \node at \weight{-5}{5} (e) {}; + \draw \weight{3}{1} -- \weight{-4}{4.5}; + \draw[dotted] \weight{-4}{4.5} -- \weight{-5}{5}; + \foreach \i in {1,...,4}{\wt[black]{5-2*\i}{\i}} + \node[above right=-2pt] at (hex cs:x=3,y=1){\small\(\lambda\)}; + \draw[-latex] (a) to[bend left=40] (b); + \draw[-latex] (b) to[bend left=40] (c); + \draw[-latex] (c) to[bend left=40] (d); + \draw[-latex] (d) to[bend left=40] (e); + \draw[-latex] (e) to[bend left=40] (d); + \draw[-latex] (d) to[bend left=40] (c); + \draw[-latex] (c) to[bend left=40] (b); + \draw[-latex] (b) to[bend left=40] (a); + \end{rootSystem} + \end{tikzpicture} +\end{center} + +What's remarkable about all this is the fact that the subalgebra spanned by +\(E_{1 2}\), \(E_{2 1}\) and \(H = [E_{1 2}, E_{2 1}]\) is isomorphic to +\(\sl_2(\CC)\) via +\begin{align*} + E_{2 1} & \mapsto e & + E_{1 2} & \mapsto f & + H & \mapsto h +\end{align*} + +In other words, \(W\) is a representation of \(\sl_2(\CC)\). Even more so, we +claim +\[ + V_{\lambda + k (\alpha_2 - \alpha_1)} = W_{\lambda(H) - 2k} +\] + +Indeed, \(V_{\lambda + k (\alpha_2 - \alpha_1)} \subset W_{\lambda(H) - 2k}\) +since \((\lambda + k (\alpha_2 - \alpha_1))(H) = \lambda(H) + k (-1 - 1) = +\lambda(H) - 2 k\). On the other hand, if we suppose \(0 < \dim V_{\lambda + k +(\alpha_2 - \alpha_1)} < \dim W_{\lambda(H) - 2 k}\) for some \(k\) we arrive +at +\[ + \dim W + = \sum_k \dim V_{\lambda + k (\alpha_2 - \alpha_1)} + < \sum_k \dim W_{\lambda(H) - 2k} + = \dim W, +\] +a contradiction. + +There are a number of important consequences to this, of the first being that +the weights of \(V\) appearing on \(W\) must be symmetric with respect to the +the line \(\langle \alpha_1 - \alpha_2, \alpha \rangle = 0\). The picture is +thus +\begin{center} + \begin{tikzpicture} + \AutoSizeWeightLatticefalse + \begin{rootSystem}{A} + \setlength{\weightRadius}{2pt} + \weightLattice{4} + \draw[thick] \weight{3}{1} -- \weight{-3}{4}; + \wt[black]{0}{0} + \node[above left] at \weight{0}{0} {\small\(0\)}; + \foreach \i in {1,...,4}{\wt[black]{5-2*\i}{\i}} + \node[above right=-2pt] at (hex cs:x=3,y=1){\small\(\lambda\)}; + \draw[very thick] \weight{0}{-4} -- \weight{0}{4} + node[above]{\small\(\langle \alpha_1 - \alpha_2, \alpha \rangle=0\)}; + \end{rootSystem} + \end{tikzpicture} +\end{center} + +Notice we could apply this same argument to the subspace \(\bigoplus_k +V_{\lambda + k (\alpha_3 - \alpha_2)}\): this subspace is invariant under the +action of the subalgebra spanned by \(E_{2 3}\), \(E_{3 2}\) and \([E_{2 3}, +E_{3 2}]\), which is again isomorphic to \(\sl_2\), so that the weights in this +subspace must be symmetric with respect to the line \(\langle \alpha_3 - +\alpha_2, \alpha \rangle = 0\). The picture is now +\begin{center} + \begin{tikzpicture} + \AutoSizeWeightLatticefalse + \begin{rootSystem}{A} + \setlength{\weightRadius}{2pt} + \weightLattice{4} + \draw[thick] \weight{3}{1} -- \weight{-3}{4}; + \draw[thick] \weight{3}{1} -- \weight{4}{-1}; + \wt[black]{0}{0} + \wt[black]{4}{-1} + \node[above left] at \weight{0}{0} {\small\(0\)}; + \foreach \i in {1,...,4}{\wt[black]{5-2*\i}{\i}} + \node[above right=-2pt] at (hex cs:x=3,y=1){\small\(\lambda\)}; + \draw[very thick] \weight{0}{-4} -- \weight{0}{4} + node[above]{\small\(\langle \alpha_1 - \alpha_2, \alpha \rangle=0\)}; + \draw[very thick] \weight{-4}{0} -- \weight{4}{0} + node[right]{\small\(\langle \alpha_3 - \alpha_2, \alpha \rangle=0\)}; + \end{rootSystem} + \end{tikzpicture} +\end{center} + +In general, given a weight \(\mu\), the space +\[ + \bigoplus_k V_{\mu + k (\alpha_i - \alpha_j)} +\] +is invariant under the action of the subalgebra \(\mathfrak{s}_{\alpha_i - +\alpha_j} = \CC \langle E_{i j}, E_{j i}, [E_{i j}, E_{j i}] \rangle\), which +is once more isomorphic to \(\sl_2(\CC)\), and again the weight spaces in this +string match precisely the eigenvalues of \(h\). Needless to say, we could keep +applying this method to the weights at the ends of our string, arriving at +\begin{center} + \begin{tikzpicture} + \AutoSizeWeightLatticefalse + \begin{rootSystem}{A} + \setlength{\weightRadius}{2pt} + \weightLattice{5} + \draw[thick] \weight{3}{1} -- \weight{-3}{4}; + \draw[thick] \weight{3}{1} -- \weight{4}{-1}; + \draw[thick] \weight{-3}{4} -- \weight{-4}{3}; + \draw[thick] \weight{-4}{3} -- \weight{-1}{-3}; + \draw[thick] \weight{1}{-4} -- \weight{4}{-1}; + \draw[thick] \weight{-1}{-3} -- \weight{1}{-4}; + \wt[black]{-4}{3} + \wt[black]{-3}{1} + \wt[black]{-2}{-1} + \wt[black]{-1}{-3} + \wt[black]{1}{-4} + \wt[black]{2}{-3} + \wt[black]{3}{-2} + \wt[black]{4}{-1} + \foreach \i in {1,...,4}{\wt[black]{5-2*\i}{\i}} + \node[above right=-2pt] at (hex cs:x=3,y=1){\small\(\lambda\)}; + \draw[very thick] \weight{-5}{5} -- \weight{5}{-5}; + \draw[very thick] \weight{0}{-5} -- \weight{0}{5}; + \draw[very thick] \weight{-5}{0} -- \weight{5}{0}; + \end{rootSystem} + \end{tikzpicture} +\end{center} + +We claim all dots \(\mu\) lying inside the hexagon we've drawn must also be +weights -- i.e. \(V_\mu \ne 0\). Indeed, by applying the same argument to an +arbitrary weight \(\nu\) in the boundary of the hexagon we get a representation +of \(\sl_2(\CC)\) whose weights correspond to weights of \(V\) lying in a +string inside the hexagon, and whose right-most weight is precisely the weight +of \(V\) we started with. +\begin{center} + \begin{tikzpicture} + \AutoSizeWeightLatticefalse + \begin{rootSystem}{A} + \setlength{\weightRadius}{2pt} + \weightLattice{5} + \draw[thick] \weight{3}{1} -- \weight{-3}{4}; + \draw[thick] \weight{3}{1} -- \weight{4}{-1}; + \draw[thick] \weight{-3}{4} -- \weight{-4}{3}; + \draw[thick] \weight{-4}{3} -- \weight{-1}{-3}; + \draw[thick] \weight{1}{-4} -- \weight{4}{-1}; + \draw[thick] \weight{-1}{-3} -- \weight{1}{-4}; + \wt[black]{-4}{3} + \wt[black]{-3}{1} + \wt[black]{-2}{-1} + \wt[black]{-1}{-3} + \wt[black]{1}{-4} + \wt[black]{2}{-3} + \wt[black]{3}{-2} + \wt[black]{4}{-1} + \foreach \i in {1,...,4}{\wt[black]{5-2*\i}{\i}} + \node[above right=-2pt] at \weight{1}{2} {\small\(\nu\)}; + \node[above right=-2pt] at (hex cs:x=3,y=1){\small\(\lambda\)}; + \draw[very thick] \weight{-5}{5} -- \weight{5}{-5}; + \draw[very thick] \weight{0}{-5} -- \weight{0}{5}; + \draw[very thick] \weight{-5}{0} -- \weight{5}{0}; + \draw[gray, thick] \weight{1}{2} -- \weight{-2}{-1}; + \wt[black]{1}{2} + \wt[black]{-2}{-1} + \wt{0}{1} + \wt{-1}{0} + \end{rootSystem} + \end{tikzpicture} +\end{center} + +By construction, \(\nu\) corresponds to the right-most weight of the +representation of \(\sl_2(\CC)\), so that all dots lying on the gray string +must occur in the representation of \(\sl_2(\CC)\). Hence they must also be +weights of \(V\). The final picture is thus +\begin{center} + \begin{tikzpicture} + \AutoSizeWeightLatticefalse + \begin{rootSystem}{A} + \setlength{\weightRadius}{2pt} + \weightLattice{5} + \draw[thick] \weight{3}{1} -- \weight{-3}{4}; + \draw[thick] \weight{3}{1} -- \weight{4}{-1}; + \draw[thick] \weight{-3}{4} -- \weight{-4}{3}; + \draw[thick] \weight{-4}{3} -- \weight{-1}{-3}; + \draw[thick] \weight{1}{-4} -- \weight{4}{-1}; + \draw[thick] \weight{-1}{-3} -- \weight{1}{-4}; + \wt[black]{-4}{3} + \wt[black]{-3}{1} + \wt[black]{-2}{-1} + \wt[black]{-1}{-3} + \wt[black]{1}{-4} + \wt[black]{2}{-3} + \wt[black]{3}{-2} + \wt[black]{4}{-1} + \foreach \i in {1,...,4}{\wt[black]{5-2*\i}{\i}} + \node[above right=-2pt] at (hex cs:x=3,y=1){\small\(\lambda\)}; + \draw[very thick] \weight{-5}{5} -- \weight{5}{-5}; + \draw[very thick] \weight{0}{-5} -- \weight{0}{5}; + \draw[very thick] \weight{-5}{0} -- \weight{5}{0}; + \wt[black]{-2}{2} + \wt[black]{0}{1} + \wt[black]{-1}{0} + \wt[black]{0}{-2} + \wt[black]{1}{-1} + \wt[black]{2}{0} + \end{rootSystem} + \end{tikzpicture} +\end{center} + +Another important consequence of our analysis is the fact that \(\lambda\) lies +in the lattice \(P\) generated by \(\alpha_1\), \(\alpha_2\) and \(\alpha_3\). +Indeed, \(\lambda([E_{i j}, E_{j i}])\) is an eigenvalue of \(h\) in a +representation of \(\sl_2(\CC)\), so it must be an integer. Now since +\[ + \lambda + \begin{pmatrix} + a & 0 & 0 \\ + 0 & b & 0 \\ + 0 & 0 & -a -b + \end{pmatrix} + = + \lambda + \begin{pmatrix} + a & 0 & 0 \\ + 0 & 0 & 0 \\ + 0 & 0 & -a + \end{pmatrix} + + + \lambda + \begin{pmatrix} + 0 & 0 & 0 \\ + 0 & b & 0 \\ + 0 & 0 & -b + \end{pmatrix} + = + a \lambda([E_{1 3}, E_{3 1}]) + b \lambda([E_{2 3}, E_{3 2}]), +\] +which is to say \(\lambda = \lambda([E_{1 3}, E_{3 1}]) \alpha_1 + +\lambda([E_{2 3}, E_{3 2}]) \alpha_2\), we can see that \(\lambda \in +P\). + +\begin{definition} + The lattice \(P = \ZZ \alpha_1 \oplus \ZZ \alpha_2 \oplus \ZZ \alpha_3\) is + called \emph{the weight lattice of \(\sl_3(\CC)\)}. +\end{definition} + +Finally\dots + +\begin{theorem}\label{thm:sl3-irr-weights-class} + The weights of \(V\) are precisely the elements of the weight lattice \(P\) + congruent to \(\lambda\) module the sublattice \(Q\) and lying inside hexagon + with vertices the images of \(\lambda\) under the group generated by + reflections across the lines \(\langle \alpha_i - \alpha_j, \alpha \rangle = + 0\). +\end{theorem} + +Once more there's a clear parallel between the case of \(\sl_3(\CC)\) and that +of \(\sl_2(\CC)\), where we observed that the weights all lied in the lattice +\(P = \ZZ\) and were congruent modulo the lattice \(Q = 2 \ZZ\). +Having found all of the weights of \(V\), the only thing we're missing is an +existence and uniqueness theorem analogous to +theorem~\ref{thm:sl2-exist-unique}. In other words, our next goal is +establishing\dots + +\begin{theorem}\label{thm:sl3-existence-uniqueness} + For each pair of positive integers \(n\) and \(m\), there exists precisely + one irreducible representation \(V\) of \(\sl_3(\CC)\) whose highest weight + is \(n \alpha_1 - m \alpha_3\). +\end{theorem} + +To proceed further we once again refer to the approach we employed in the case +of \(\sl_2(\CC)\): next we showed in theorem~\ref{thm:basis-of-irr-rep} that +any irreducible representation of \(\sl_2(\CC)\) is spanned by the images of +its highest weight vector under \(f\). A more abstract way of putting it is to +say that an irreducible representation \(V\) of \(\sl_2(\CC)\) is spanned by +the images of its highest weight vector under successive applications by half +of the root spaces of \(\sl_2(\CC)\). The advantage of this alternative +formulation is, of course, that the same holds for \(\sl_3(\CC)\). +Specifically\dots + +\begin{theorem}\label{thm:irr-sl3-span} + Given an irreducible \(\sl_3(\CC)\)-representation \(V\) and a highest + weight vector \(v \in V\), \(V\) is spanned by the images of \(v\) under + successive applications of \(E_{2 1}\), \(E_{3 1}\) and \(E_{3 2}\). +\end{theorem} + +The proof of theorem~\ref{thm:irr-sl3-span} is very similar to that of +theorem~\ref{thm:basis-of-irr-rep}: we use the commutator relations of +\(\sl_3(\CC)\) to inductively show that the subspace spanned by the images of a +highest weight vector under successive applications of \(E_{2 1}\), \(E_{3 1}\) +and \(E_{3 2}\) is invariant under the action of \(\sl_3(\CC)\) -- please refer +to \cite{fulton-harris} for further details. The same argument also goes to +show\dots + +\begin{corollary} + Given a representation \(V\) of \(\sl_3(\CC)\) with highest weight + \(\lambda\) and \(v \in V_\lambda\), the subspace spanned by successive + applications of \(E_{2 1}\), \(E_{3 1}\) and \(E_{3 2}\) to \(v\) is an + irreducible subrepresentation whose highest weight is \(\lambda\). +\end{corollary} + +This is very interesting to us since it implies that finding \emph{any} +representation whose highest weight is \(n \alpha_1 - m \alpha_2\) is enough +for establishing the ``existence'' part of +theorem~\ref{thm:sl3-existence-uniqueness}. Moreover, constructing such +representation turns out to be quite simple. + +\begin{proof}[Proof of existence] + Consider the natural representation \(V = \CC^3\) of \(\sl_3(\CC)\). We + claim that the highest weight of \(\Sym^n V \otimes \Sym^m V^*\) + is \(n \alpha_1 - m \alpha_3\). + + First of all, notice that the eigenvectors of \(V\) are the canonical basis + vectors \(e_1\), \(e_2\) and \(e_3\), whose eigenvalues are \(\alpha_1\), + \(\alpha_2\) and \(\alpha_3\) respectively. Hence the weight diagram of \(V\) + is + \begin{center} + \begin{tikzpicture}[scale=2.5] + \AutoSizeWeightLatticefalse + \begin{rootSystem}{A} + \weightLattice{2} + \wt[black]{1}{0} + \wt[black]{-1}{1} + \wt[black]{0}{-1} + \node[right] at \weight{1}{0} {$\alpha_1$}; + \node[above left] at \weight{-1}{1} {$\alpha_2$}; + \node[below left] at \weight{0}{-1} {$\alpha_3$}; + \end{rootSystem} + \end{tikzpicture} + \end{center} + and \(\alpha_1\) is the highest weight of \(V\). + + On the one hand, if \(\{f_1, f_2, f_3\}\) is the dual basis of \(\{e_1, e_2, + e_3\}\) then \(H f_i = - \alpha_i(H) \cdot f_i\) for each \(H \in + \mathfrak{h}\), so that the weights of \(V^*\) are precisely the opposites of + the weights of \(V\). In other words, + \begin{center} + \begin{tikzpicture}[scale=2.5] + \AutoSizeWeightLatticefalse + \begin{rootSystem}{A} + \weightLattice{2} + \wt[black]{-1}{0} + \wt[black]{1}{-1} + \wt[black]{0}{1} + \node[left] at \weight{-1}{0} {$-\alpha_1$}; + \node[below right] at \weight{1}{-1} {$-\alpha_2$}; + \node[above right] at \weight{0}{1} {$-\alpha_3$}; + \end{rootSystem} + \end{tikzpicture} + \end{center} + is the weight diagram of \(V^*\) and \(\alpha_3\) is the highest weight of + \(V^*\). + + On the other hand if we fix two \(\sl_3(\CC)\)-representations \(U\) and + \(W\), by computing + \[ + \begin{split} + H (u \otimes w) + & = H u \otimes w + u \otimes H w \\ + & = \lambda(H) \cdot u \otimes w + u \otimes \mu(H) \cdot w \\ + & = (\lambda + \mu)(H) \cdot (u \otimes w) + \end{split} + \] + for each \(H \in \mathfrak{h}\), \(u \in U_\lambda\) and \(w \in W_\lambda\) + we can see that the weights of \(U \otimes W\) are precisely the sums of the + weights of \(U\) with the weights of \(W\). + + This implies that the maximal weights of \(\Sym^n V\) and \(\Sym^m V^*\) are + \(n \alpha_1\) and \(- m \alpha_3\) respectively -- with maximal weight + vectors \(e_1^n\) and \(f_3^m\). Furthermore, by the same token the highest + weight of \(\Sym^n V \otimes \Sym^m V^*\) must be \(n e_1 - m e_3\) -- with + highest weight vector \(e_1^n \otimes f_3^m\). +\end{proof} + +The ``uniqueness'' part of theorem~\ref{thm:sl3-existence-uniqueness} is even +simpler than that. + +\begin{proof}[Proof of uniqueness] + Let \(V\) and \(W\) be two irreducible representations of \(\sl_3(\CC)\) with + highest weight \(\lambda\). By theorem~\ref{thm:sl3-irr-weights-class}, the + weights of \(V\) are precisely the same as those of \(W\). + + Now by computing + \[ + H (v + w) + = H v + H w + = \mu(H) \cdot v + \mu(H) \cdot w + = \mu(H) \cdot (v + w) + \] + for each \(H \in \mathfrak{h}\), \(v \in V_\mu\) and \(w \in W_\mu\), we can + see that the weights of \(V \oplus W\) are same as those of \(V\) and \(W\). + Hence the highest weight of \(V \oplus W\) is \(\lambda\) -- with highest + weight vectors given by the sum of highest weight vectors of \(V\) and \(W\). + + Fix some \(v \in V_\lambda\) and \(w \in W_\lambda\) and consider the + irreducible representation \(U = \sl_3(\CC) \ v + w\) generated by \(v + w\). + The projection maps \(\pi_1 : U \to V\), \(\pi_2 : U \to W\), being non-zero + homomorphism between irreducible representations of \(\sl_3(\CC)\) must be + isomorphism. Finally, + \[ + V \cong U \cong W + \] +\end{proof} + +The situation here is analogous to that of the previous section, where we saw +that the irreducible representations of \(\sl_2(\CC)\) are given by symmetric +powers of the natural representation. + +We've been very successful in our pursue for a classification of the +irreducible representations of \(\sl_2(\CC)\) and \(\sl_3(\CC)\), but so far +we've mostly postponed the discussion on the motivation behind our methods. In +particular, we did not explain why we chose \(h\) and \(\mathfrak{h}\), and +neither why we chose to look at their eigenvalues. Apart from the obvious fact +we already knew it would work a priory, why did we do all that? In the +following section we will attempt to answer this question by looking at what we +did in the last chapter through more abstract lenses and studying the +representations of an arbitrary finite-dimensional complex semisimple Lie +algebra \(\mathfrak{g}\). + +\section{Simultaneous Diagonalization \& the General Case} + +At the heart of our analysis of \(\sl_2(\CC)\) and \(\sl_3(\CC)\) was the +decision to consider the eigenspace decomposition +\begin{equation}\label{sym-diag} + V = \bigoplus_\lambda V_\lambda +\end{equation} + +This was simple enough to do in the case of \(\sl_2(\CC)\), but the reasoning +behind it, as well as the mere fact equation (\ref{sym-diag}) holds, are harder +to explain in the case of \(\sl_3(\CC)\). The eigenspace decomposition +associated with an operator \(V \to V\) is a very well-known tool, and this +type of argument should be familiar to anyone familiar with basic concepts of +linear algebra. On the other hand, the eigenspace decomposition of \(V\) with +respect to the action of an arbitrary subalgebra \(\mathfrak{h} \subset +\gl(V)\) is neither well-known nor does it hold in general: as previously +stated, it may very well be that +\[ + \bigoplus_{\lambda \in \mathfrak{h}^*} V_\lambda \subsetneq V +\] + +We should note, however, that this two cases are not as different as they may +sound at first glance. Specifically, we can regard the eigenspace decomposition +of a representation \(V\) of \(\sl_2(\CC)\) with respect to the eigenvalues of +the action of \(h\) as the eigenvalue decomposition of \(V\) with respect to +the action of the subalgebra \(\mathfrak{h} = \CC h \subset \sl_2(\CC)\). +Furthermore, in both cases \(\mathfrak{h} \subset \sl_n(\CC)\) is the +subalgebra of diagonal matrices, which is Abelian. The fundamental difference +between these two cases is thus the fact that \(\dim \mathfrak{h} = 1\) for +\(\mathfrak{h} \subset \sl_2(\CC)\) while \(\dim \mathfrak{h} > 1\) for +\(\mathfrak{h} \subset \sl_3(\CC)\). The question then is: why did we choose +\(\mathfrak{h}\) with \(\dim \mathfrak{h} > 1\) for \(\sl_3(\CC)\)? + +The rational behind fixing an Abelian subalgebra is one we have already +encountered when dealing with finite groups: representations of Abelian groups +and algebras are generally much simpler to understand than the general case. +Thus it make sense to decompose a given representation \(V\) of +\(\mathfrak{g}\) into subspaces invariant under the action of \(\mathfrak{h}\), +and then analyze how the remaining elements of \(\mathfrak{g}\) act on this +subspaces. The bigger \(\mathfrak{h}\) the simpler our problem gets, because +there are fewer elements outside of \(\mathfrak{h}\) left to analyze. + +Hence we are generally interested in maximal Abelian subalgebras \(\mathfrak{h} +\subset \mathfrak{g}\). When \(\mathfrak{g}\) is semisimple, these coincide +with the so called \emph{Cartan subalgebras} of \(\mathfrak{g}\) -- i.e. +self-normalizing nilpotent subalgebras. A simple argument via Zorn's lemma is +enough to establish the existence of Cartan subalgebras for semisimple +\(\mathfrak{g}\): it suffices to note that if +\[ + \mathfrak{h}_1 + \subset \mathfrak{h}_2 + \subset \cdots + \subset \mathfrak{h}_n + \subset \cdots +\] +is a chain of Abelian subalgebras, then their sum is again an Abelian +subalgebra. Alternatively, one can show that every compact Lie group \(G\) +contains a maximal tori, whose Lie algebra is therefore a maximal Abelian +subalgebra of \(\mathfrak{g}\). + +That said, we can easily compute concrete examples. For instance, one can +readily check that every pair of diagonal matrices commutes, so that +\[ + \mathfrak{h} = + \begin{pmatrix} + \CC & 0 & \cdots & 0 \\ + 0 & \CC & \cdots & 0 \\ + \vdots & \vdots & \ddots & \vdots \\ + 0 & 0 & \cdots & \CC + \end{pmatrix} +\] +is an Abelian subalgebra of \(\gl_n(\CC)\). A simple calculation then shows +that if \(X \in \gl_n(\CC)\) commutes with every diagonal matrix \(H \in +\mathfrak{h}\) then \(X\) is a diagonal matrix, so that \(\mathfrak{h}\) is a +Cartan subalgebra of \(\gl_n(\CC)\). The intersection of such subalgebra with +\(\sl_n(\CC)\) -- i.e. the subalgebra of traceless diagonal matrices -- is a +Cartan subalgebra of \(\sl_n(\CC)\). In particular, if \(n = 2\) or \(n = 3\) +we get to the subalgebras described the previous two sections. + +The remaining question then is: if \(\mathfrak{h} \subset \mathfrak{g}\) is a +Cartan subalgebra and \(V\) is a representation of \(\mathfrak{g}\), does the +eigenspace decomposition +\[ + V = \bigoplus_{\lambda \in \mathfrak{h}^*} V_\lambda +\] +of \(V\) hold? The answer to this question turns out to be yes. This is a +consequence of something known as \emph{simultaneous diagonalization}, which is +the primary tool we'll use to generalize the results of the previous section. +What is simultaneous diagonalization all about then? + +\begin{proposition} + Let \(\mathfrak{g}\) be a Lie algebra, \(\mathfrak{h} \subset \mathfrak{g}\) + be an Abelian subalgebra and \(V\) be any finite-dimensional representation + of \(\mathfrak{g}\). Then there is a basis \(\{v_1, \ldots, v_n\}\) of \(V\) + so that each \(v_i\) is simultaneously an eigenvector of all elements of + \(\mathfrak{h}\) -- i.e. each element of \(\mathfrak{h}\) acts as a diagonal + matrix in this basis. In other words, there is some linear functional + \(\lambda \in \mathfrak{h}^*\) so that + \[ + H v_i = \lambda(H) \cdot v_i + \] + for all \(H \in \mathfrak{h}\) and all \(i\). +\end{proposition} + +\begin{proof} + We claim \(\mathfrak{h}\) is semisimple. Indeed, if \(\{H_1, \ldots, H_m\}\) + is basis of \(\mathfrak{h}\) then + \[ + \mathfrak{h} \cong \bigoplus_i \CC H_i + \] + as vector spaces. Usually this is simply a linear isomorphism, but since + \(\mathfrak{h}\) is Abelian this is an isomorphism of Lie algebras -- here + \(\CC H_i\) represents the 1-dimensional subalgebra spanned by \(H_i\), which + is isomorphic to the trivial Lie algebra \(\CC\). Each \(\CC H_i\) is simple, + so \(\mathfrak{h}\) is isomorphic to a direct sum of simple algebras -- i.e. + \(\mathfrak{h}\) is semisimple. + + Hence + \[ + V + = \Res{\mathfrak g}{\mathfrak h} V + \cong \bigoplus V_i, + \] + as representations of \(\mathfrak{h}\), where each \(V_i\) is an irreducible + representation of \(\mathfrak{h}\). Since \(\mathfrak{h}\) is Abelian, it + follows from Schur's lemma that each \(V_i\) is 1-dimensional. Say \(V_i = + \CC v_i\) and consider the basis \(\{v_1, \ldots, v_n\}\) of \(V\). Now the + assertion that each \(v_i\) is an eigenvector of all elements of + \(\mathfrak{h}\) is equivalent to the statement that each \(\CC v_i\) is + stable under the action of \(\mathfrak{h}\). +\end{proof} + +As promised, this implies\dots + +\begin{corollary} + Let \(\mathfrak{g}\) be a finite-dimensional complex semisimple Lie algebra + and \(\mathfrak{h}\) be a Cartan subalgebra of \(\mathfrak{g}\). Given a + finite-dimensional representation \(V\) of \(\mathfrak{g}\), + \[ + V = \bigoplus_{\lambda \in \mathfrak{h}^*} V_\lambda + \] +\end{corollary} + +We now have most of the necessary tools to reproduce the results of the +previous chapter in a general setting. Let \(\mathfrak{g}\) be a +finite-dimensional semisimple algebra with a Cartan subalgebra \(\mathfrak{h}\) +and let \(V\) be a finite-dimensional irreducible representation of +\(\mathfrak{g}\). We will proceed, as we did before, by generalizing the +results about of the previous two sections in order. By now the pattern should +be starting become clear, so we will mostly omit technical details and proofs +analogous to the ones on the previous sections. Further details can be found in +appendix D of \cite{fulton-harris} and in \cite{humphreys}. + +We begin our analysis by remarking that in both \(\sl_2(\CC)\) and +\(\sl_3(\CC)\), the roots were symmetric about the origin and spanned all of +\(\mathfrak{h}^*\). This turns out to be a general fact, which is a consequence +of the following theorem. + +% TODO: Add a proof? The proof of FH turns out to be recursive!!!! +% TODO: Note that this is where the maximality of the Cartan subalgebra comes +% into play +\begin{theorem} + If \(\mathfrak g\) is semisimple then its Killing form \(K\) is + non-degenerate. Furthermore, the restriction of \(K\) to \(\mathfrak{h}\) is + non-degenerate. +\end{theorem} + +\begin{proposition}\label{thm:weights-symmetric-span} + The eigenvalues \(\alpha\) of the adjoint action of \(\mathfrak{h}\) in + \(\mathfrak{g}\) are symmetrical about the origin -- i.e. \(- \alpha\) is + also an eigenvalue -- and they span all of \(\mathfrak{h}^*\). +\end{proposition} + +\begin{proof} + We'll start with the first claim. Let \(\alpha\) and \(\beta\) be two + eigenvalues of the adjoint action of \(\mathfrak{h}\). Notice + \([\mathfrak{g}_\alpha, \mathfrak{g}_\beta] \subset \mathfrak{g}_{\alpha + + \beta}\). Indeed, if \(X \in \mathfrak{g}_\alpha\) and \(Y \in + \mathfrak{g}_\beta\) then + \[ + [H [X, Y]] + = [X, [H, Y]] - [Y, [H, X]] + = (\alpha + \beta)(H) \cdot [X, Y] + \] + for all \(H \in \mathfrak{h}\). + + This implies that if \(\alpha + \beta \ne 0\) then \(\ad(X) \ad(Y)\) is + nilpotent: if \(Z \in \mathfrak{g}_\gamma\) then + \[ + (\ad(X) \ad(Y))^n Z + = [X, [Y, [ \ldots, [X, [Y, Z]]] \ldots ] + \in \mathfrak{g}_{n \alpha + n \beta + \gamma} + = 0 + \] + for \(n\) large enough. In particular, \(K(X, Y) = \Tr(\ad(X) \ad(Y)) = 0\). + Now if \(- \alpha\) is not an eigenvalue we find \(K(X, \mathfrak{g}_\beta) = + 0\) for all eigenvalues \(\beta\), which contradicts the non-degeneracy of + \(K\). Hence \(- \alpha\) must be an eigenvalue of the adjoint action of + \(\mathfrak{h}\). + + For the second statement, note that if the eigenvalues of \(\mathfrak{h}\) do + not span all of \(\mathfrak{h}^*\) then there is some \(H \in \mathfrak{h}\) + non-zero such that \(\alpha(H) = 0\) for all eigenvalues \(\alpha\), which is + to say, \(\ad(H) X = [H, X] = 0\) for all \(X \in \mathfrak{g}\). Another way + of putting it is to say \(H\) is an element of the center \(\mathfrak{z}\) of + \(\mathfrak{g}\), which is zero by the semisimplicity -- a contradiction. +\end{proof} + +Furthermore, as in the case of \(\sl_2(\CC)\) and \(\sl_3(\CC)\) one can +show\dots + +\begin{proposition}\label{thm:root-space-dim-1} + The eigenspaces \(\mathfrak{g}_\alpha\) are all 1-dimensional. +\end{proposition} + +The proof of the first statement of +proposition~\ref{thm:weights-symmetric-span} highlights something interesting: +if we fix some some eigenvalue \(\alpha\) of the adjoint action of +\(\mathfrak{h}\) in \(\mathfrak{g}\) and a eigenvector \(X \in +\mathfrak{g}_\alpha\), then for each \(H \in \mathfrak{h}\) and \(v \in +V_\lambda\) we find +\[ + H (X v) + = X (H v) + [H, X] v + = (\lambda + \alpha)(H) \cdot X v +\] +so that \(X\) carries \(v\) to \(V_{\lambda + \alpha}\). We have encountered +this formula twice in this chapter: again, we find \(\mathfrak{g}_\alpha\) +\emph{acts on \(V\) by translating vectors between eigenspaces}. In other +words, if we denote by \(\Delta\) the set of all roots of \(\mathfrak{g}\) +then\dots + +\begin{theorem}\label{thm:weights-congruent-mod-root} + The weights of an irreducible representation \(V\) of \(\mathfrak{g}\) are + all congruent module the root lattice \(Q = \ZZ \Delta\) of \(\mathfrak{g}\). +\end{theorem} + +To proceed further, as in the case of \(\sl_3(\CC)\) we have to fix a direction +in \(\mathfrak{h}^*\) -- i.e. we fix a linear function \(\mathfrak{h}^* \to +\RR\) such that \(Q\) lies outside of its kernel. This choice induces a +partition \(\Delta = \Delta^+ \cup \Delta^-\) of the set of roots of +\(\mathfrak{g}\) and once more we find\dots + +\begin{theorem} + There is a weight vector \(v \in V\) that is killed by all positive root + spaces of \(\mathfrak{g}\). +\end{theorem} + +\begin{proof} + It suffices to note that if \(\lambda\) is the weight of \(V\) lying the + furthest along the direction we chose and \(V_{\lambda + \alpha} \ne 0\) for + some \(\alpha \in \Delta^+\) then \(\lambda + \alpha\) is a weight that is + furthest along the direction we chose than \(\lambda\), which contradicts the + definition of \(\lambda\). +\end{proof} + +Accordingly, we call \(\lambda\) \emph{the highest weight of \(V\)}, and we +call any \(v \in V_\lambda\) \emph{a highest weight vector}. The strategy then +is to describe all weight spaces of \(V\) in terms of \(\lambda\) and \(v\), as +in theorem~\ref{thm:sl3-irr-weights-class}, and unsurprisingly we do so by +reproducing the proof of the case of \(\sl_3(\CC)\). Namely, we show\dots + +\begin{proposition}\label{thm:distinguished-subalgebra} + Given a root \(\alpha\) of \(\mathfrak{g}\) the subspace + \(\mathfrak{s}_\alpha = \mathfrak{g}_\alpha \oplus \mathfrak{g}_{- \alpha} + \oplus [\mathfrak{g}_\alpha, \mathfrak{g}_{- \alpha}]\) is a subalgebra + isomorphic to \(\sl_2(\CC)\). +\end{proposition} + +\begin{corollary}\label{thm:distinguished-subalg-rep} + For all weights \(\mu\), the subspace + \[ + V_\mu[\alpha] = \bigoplus_k V_{\mu + k \alpha} + \] + is invariant under the action of the subalgebra \(\mathfrak{s}_\alpha\) + and the weight spaces in this string match the eigenspaces of \(h\). +\end{corollary} + +The proof of proposition~\ref{thm:distinguished-subalgebra} is very technical +in nature and we won't include it here, but the idea behind it is simple: +recall that \(\mathfrak{g}_\alpha\) and \(\mathfrak{g}_{- \alpha}\) are both +1-dimensional, so that \(\dim [\mathfrak{g}_\alpha, \mathfrak{g}_{- \alpha}]\) +is at most 1. We check that \([\mathfrak{g}_\alpha, \mathfrak{g}_{- \alpha}] +\ne 0\) and that no generator of \([\mathfrak{g}_\alpha, \mathfrak{g}_{- +\alpha}] \ne 0\) is annihilated by \(\alpha\), so that by adjusting scalars we +can find \(E_\alpha \in \mathfrak{g}_\alpha\) and \(F_\alpha \in +\mathfrak{g}_{- \alpha}\) such that \(H_\alpha = [E_\alpha, F_\alpha]\) +satisfies +\begin{align*} + [H_\alpha, F_\alpha] & = -2 F_\alpha & + [H_\alpha, E_\alpha] & = 2 E_\alpha +\end{align*} + +The elements \(E_\alpha, F_\alpha \in \mathfrak{g}\) are not uniquely +determined by this condition, but \(H_\alpha\) is. The second statement of +corollary~\ref{thm:distinguished-subalg-rep} imposes a restriction on the +weights of \(V\). Namely, if \(\mu\) is a weight, \(\mu(H_\alpha)\) is an +eigenvalue of \(h\) in some representation of \(\sl_2(\CC)\), so that\dots + +\begin{proposition} + The weights \(\mu\) of an irreducible representation \(V\) of + \(\mathfrak{g}\) are so that \(\mu(H_\alpha) \in \ZZ\) for each \(\alpha \in + \Delta\). +\end{proposition} + +Once more, the lattice \(P = \{ \lambda \in \mathfrak{h}^* : \lambda(H_\alpha) +\in \ZZ, \forall \alpha \in \Delta \}\) is called \emph{the weight lattice of +\(\mathfrak{g}\)}, and we call the elements of \(P\) \emph{integral}. Finally, +another important consequence of theorem~\ref{thm:distinguished-subalgebra} +is\dots + +\begin{corollary} + If \(\alpha \in \Delta^+\) and \(T_\alpha : \mathfrak{h}^* \to + \mathfrak{h}^*\) is the reflection in the hyperplane perpendicular to + \(\alpha\) with respect to the Killing form, + corollary~\ref{thm:distinguished-subalg-rep} implies that all \(\nu \in P\) + lying inside the line connecting \(\mu\) and \(T_\alpha \mu\) are weights -- + i.e. \(V_\nu \ne 0\). +\end{corollary} + +\begin{proof} + It suffices to note that \(\nu \in V_\mu[\alpha]\) -- see appendix D of + \cite{fulton-harris} for further details. +\end{proof} + +\begin{definition} + We refer to the group \(W = \langle T_\alpha : \alpha \in \Delta^+ \rangle + \subset \operatorname{O}(\mathfrak{h}^*)\) as \emph{the Weyl group of + \(\mathfrak{g}\)}. +\end{definition} + +This is entirely analogous to the situation of \(\sl_3(\CC)\), where we found +that the weights of the irreducible representations were symmetric with respect +to the lines \(\langle \alpha_i - \alpha_j, \alpha \rangle = 0\). Indeed, the +same argument leads us to the conclusion\dots + +\begin{theorem}\label{thm:irr-weight-class} + The weights of an irreducible representation \(V\) of \(\mathfrak{g}\) with + highest weight \(\lambda\) are precisely the elements of the weight lattice + \(P\) congruent to \(\lambda\) modulo the root lattice \(Q\) lying inside the + convex hull of the image of \(\lambda\) under the action of the Weyl group + \(W\). +\end{theorem} + +Now the only thing we are missing for a complete classification is an existence +and uniqueness theorem analogous to theorem~\ref{thm:sl2-exist-unique} and +theorem~\ref{thm:sl3-existence-uniqueness}. Lo and behold\dots + +\begin{theorem}\label{thm:dominant-weight-theo} + For each \(\lambda \in P\) such that \(\lambda(H_\alpha) \ge 0\) for + all positive roots \(\alpha\) there exists precisely one irreducible + representation \(V\) of \(\mathfrak{g}\) whose highest weight is \(\lambda\). +\end{theorem} + +\begin{note} + An element \(\lambda\) of \(P\) such that \(\lambda(H_\alpha) \ge 0\) for all + \(\alpha \in \Delta^+\) is usually referred to as an \emph{integral + dominant weight of \(\mathfrak{g}\)}. +\end{note} + +Unsurprisingly, our strategy is to copy what we did in the previous section. +The ``uniqueness'' part of the theorem follows at once from the argument used +for \(\sl_3(\CC)\), and the proof of existence of can once again be reduced +to the proof of\dots + +\begin{theorem}\label{thm:weak-dominant-weight} + There exists \emph{some} -- not necessarily irreducible -- finite-dimensional + representation of \(\mathfrak{g}\) whose highest weight is \(\lambda\). +\end{theorem} + +The trouble comes when we try to generalize the proof of +theorem~\ref{thm:weak-dominant-weight} we used for the case when \(\mathfrak{g} += \sl_3(\CC)\). The issue is that our proof relied heavily on our knowledge of +the roots of \(\sl_3(\CC)\). Specifically, we used the fact every dominant +integral weight of \(\sl_3(\CC)\) can be written as \(n \alpha_1 - m \alpha_3\) +for unique non-negative integers \(n\) and \(m\). When then constructed +finite-dimensional representations \(V\) and \(W\) of \(\sl_3(\CC)\) whose +highest weights are \(\alpha_1\) and \(- \alpha_3\), so that the highest weight +of \(\Sym^n V \otimes \Sym^m W\) is \(n \alpha_1 - m \alpha_3\). + +A similar construction can be implemented for \(\sl_n(\CC)\): if \(\mathfrak{h} +\subset \sl_n(\CC)\) is the subalgebra of diagonal matrices -- which, as you +may recall, is a Cartan subalgebra -- and \(\alpha : \mathfrak{h} \to \CC\) is +given by \(\alpha(E_{j j}) = \delta_{i j}\), one can show that any dominant +integral weight of \(\sl_n(\CC)\) can be uniquely expressed in the form \(k_1 +\alpha_1 + k_2 (\alpha_1 + \alpha_2) + \cdots + k_{n - 1} (\alpha_1 + \cdots + +\alpha_{n - 1})\) for non-negative integers \(k_1, k_2, \ldots k_{n - 1}\). For +instance, one may visually represent the roots of \(\sl_4(\CC)\) by +\begin{center} + \begin{tikzpicture}[scale=3] + \draw (0, 0) -- (1, 0) -- (1, 1) -- (0, 1) -- cycle; + \draw (0, 1) -- (.4, 1.4) -- (1.4, 1.4) -- (1, 1); + \draw (1, 0) -- (1.4, .4) -- (1.4, 1.4); + \draw[dotted] (0, 0) -- (.4, .4) -- (1.4, .4); + \draw[dotted] (.4, 1.4) -- (.4, .4); + + \filldraw (.5, 0) circle (.7pt); + \filldraw (.5, 1) circle (.7pt); + \node[below] at (.5, 0) {$\alpha_2 - \alpha_1$}; + + \filldraw ( .4, .9) circle (.7pt); + \filldraw (1.4, .9) circle (.7pt); + \node[right] at (1.4, .9) {$\alpha_1 - \alpha_4$}; + + \filldraw (.9, .4) circle (.7pt); + \filldraw (.9, 1.4) circle (.7pt); + \node[above] at (.9, 1.4) {$\alpha_1 - \alpha_2$}; + + \filldraw (0, .5) circle (.7pt); + \filldraw (1, .5) circle (.7pt); + \node[left] at (0, .5) {$\alpha_4 - \alpha_1$}; + + \filldraw (.2, .2) circle (.7pt); + \filldraw (.2, 1.2) circle (.7pt); + \node[above left] at (.2, 1.2) {$\alpha_4 - \alpha_2$}; + + \filldraw (1.2, .2) circle (.7pt); + \filldraw (1.2, 1.2) circle (.7pt); + \node[below right] at (1.2, .2) {$\alpha_2 - \alpha_4$}; + \end{tikzpicture} +\end{center} + +% TODO: Historical citation needed! +% TODO: Mention at the start of the chapter that we are following Weyl's +% footsteps in here +One can then construct representations \(V_i\) of \(\sl_n(\CC)\) whose highest +weights are \(\alpha_1 + \cdots + \alpha_i\). In fact, whenever we can find +finitely many generators of \(\beta_i\) of the set of dominant integral weights +and finite-dimensional representations \(V_i\) of \(\mathfrak{g}\) whose +highest weights are \(\beta_i\) we can construct a finite-dimensional +representation of \(\mathfrak{g}\) whose highest weight is some dominant +integral \(\lambda \in P\) by tensoring symmetric powers of the \(V_i\)'s. This +is the approach we'll take to prove theorem~\ref{thm:weak-dominant-weight}, as +historically this was Weyl's first proof of the theorem. As of now, however, we +don't have the necessary tools to construct a standard set of generators of the +dominant integral weights of some arbitrary semisimple \(\mathfrak{g}\), let +alone the representations \(V_i\). Indeed, Weyl's work was based on Cartan's +classification of finite-dimensional complex simple Lie algebras, which we so +far have neglected to mention. + +Alternatively, one could construct a potentially infinite-dimensional +representation of \(\mathfrak{g}\) whose highest weight is some fixed dominant +integral weight \(\lambda\) by taking the induced representation +\(\Ind{\mathfrak{g}}{\mathfrak{b}} V_\lambda = \mathcal{U}(\mathfrak{g}) +\otimes_{\mathcal{U}(\mathfrak{b})} V_\lambda\), where \(\mathfrak{b} = +\mathfrak{h} \oplus \bigoplus_{\alpha \in \Delta^+} \mathfrak{g}_\alpha \subset +\mathfrak{g}\) is the so called \emph{Borel subalgebra of \(\mathfrak{g}\)}, +\(\mathcal{U}(\mathfrak{g})\) denotes the \emph{universal enveloping algebra +of \(\mathfrak{g}\)} and \(\mathfrak{b}\) acts on \(V_\lambda = \CC v\) via \(H +v = \lambda(H) \cdot v\) and \(X v = 0\) for \(X \in \mathfrak{g}_\alpha\), as +does \cite{humphreys} in his proof. The fact that \(v\) is annihilated by all +positive root spaces guarantees that the maximal weight of \(V\) is at most +\(\lambda\), while the Poincare-Birkhoff-Witt \cite{humphreys} theorem +guarantees that \(v = 1 \otimes v \in V\) is a non-zero weight vector of +\(\lambda\) -- so that \(\lambda\) is the highest weight of \(V\). +The challenge then is to show that the irreducible component of \(v\) in \(V\) +is finite-dimensional -- see chapter 20 of \cite{humphreys} for a proof. + +This approach has the advantage of working over fields other than \(\CC\), but +in keeping with our general theme of preferring geometric proofs over purely +algebraic ones we will instead take this as an opportunity to dive into +Cartan's classification. In the next chapter we will explore the structure of +complex semisimple Lie algebras, and in the process of doing so we will reduce +the proof of theorem~\ref{thm:weak-dominant-weight} to a proof by exhaustion. +
diff --git a/sections/preface.tex b/sections/preface.tex @@ -0,0 +1,57 @@ +\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. + +Throughout this notes we'll follow the following guiding principles: + +\begin{enumerate} + \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 + by other materials. Otherwise we refer the reader to the proofs in other + books or articles -- which will likely be indicated just after the + 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. +
diff --git a/tcc.tex b/tcc.tex @@ -0,0 +1,32 @@ +\input{./preamble.tex} +\addbibresource{references.bib} + +\title{Representation Theory} +\subtitle{The Algebraic \& the Geometric} +\author{Thiago Brevidelli Garcia} + +\begin{document} + +\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. +\end{dedication} + +\pagenumbering{roman} +\setcounter{page}{1} + +\input{sections/preface} + +\tableofcontents + +\pagenumbering{arabic} +\setcounter{page}{1} + +\input{sections/lie-algebras} + +\printbibliography +\end{document} +