\documentclass[10pt,notitlepage]{article} \usepackage{amsmath, graphicx, amssymb, stmaryrd, datetime, multicol, calc, import, amscd, picins, enumitem, needspace, import} %\usepackage{fdsymbol} % For \coloneqq. %\usepackage{wasysym} %\usepackage{txfonts} % For \coloneqq; but harms \calA. %\usepackage{pxfonts} % For \multimapdotboth \usepackage{mathtools} % For \coloneqq. \usepackage{mathbbol} % For \bbe; sometimes harmed by later packages. \usepackage[usenames,dvipsnames]{xcolor} % Following http://tex.stackexchange.com/a/847/22475: \usepackage[setpagesize=false]{hyperref}\hypersetup{colorlinks, linkcolor={green!50!black}, citecolor={green!50!black}, urlcolor=blue } \usepackage[all]{xy} \usepackage{pstricks} %\usepackage[greek,english]{babel} \newcommand\entry[1]{{\bf\tiny (#1)}} % Following http://tex.stackexchange.com/questions/23521/tabular-vertical-alignment-to-top: \def\imagetop#1{\vtop{\null\hbox{#1}}} \def\red{\color{red}} \def\greenm#1{{\setlength{\fboxsep}{0pt}\colorbox{LimeGreen}{$#1$}}} \def\greent#1{{\setlength{\fboxsep}{0pt}\colorbox{LimeGreen}{#1}}} \def\pinkm#1{{\setlength{\fboxsep}{0pt}\colorbox{pink}{$#1$}}} \def\pinkt#1{{\setlength{\fboxsep}{0pt}\colorbox{pink}{#1}}} \def\purplem#1{{\setlength{\fboxsep}{0pt}\colorbox{Thistle}{$#1$}}} \def\purplet#1{{\setlength{\fboxsep}{0pt}\colorbox{Thistle}{#1}}} \def\yellowm#1{{\setlength{\fboxsep}{0pt}\colorbox{yellow}{$#1$}}} \def\yellowt#1{{\setlength{\fboxsep}{0pt}\colorbox{yellow}{#1}}} \def\ds{\displaystyle} \newcommand{\Ad}{\operatorname{Ad}} \newcommand{\ad}{\operatorname{ad}} \newcommand{\bch}{\operatorname{bch}} \newcommand{\der}{\operatorname{der}} \newcommand{\diver}{\operatorname{div}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\mor}{\operatorname{mor}} \def\sder{\operatorname{\mathfrak{sder}}} \def\barT{{\bar T}} \def\bbe{\mathbb{e}} \def\bbD{{\mathbb D}} \def\bbE{{\mathbb E}} \def\bbO{{\mathbb O}} \def\bbQ{{\mathbb Q}} \def\bbZ{{\mathbb Z}} \def\bcA{{\bar{\mathcal A}}} \def\calA{{\mathcal A}} \def\calD{{\mathcal D}} \def\calF{{\mathcal F}} \def\calG{{\mathcal G}} \def\calL{{\mathcal L}} \def\calO{{\mathcal O}} \def\calP{{\mathcal P}} \def\calS{{\mathcal S}} \def\calU{{\mathcal U}} \def\frakb{{\mathfrak b}} \def\frakg{{\mathfrak g}} \def\frakt{{\mathfrak t}} \def\tilq{{\tilde{q}}} \def\act{{\hspace{-1pt}\sslash\hspace{-0.75pt}}} \def\CYB{\operatorname{CYB}} \def\d{\downarrow} \def\dd{{\downarrow\downarrow}} \def\e{\epsilon} \def\CW{\text{\it CW}} \def\FA{\text{\it FA}} \def\FL{\text{\it FL}} \def\Loneco{{\calL^{\text{1co}}}} \def\PaT{\text{{\bf PaT}}} \def\PvT{{\mathit P\!v\!T}} \def\remove{\!\setminus\!} \def\SW{\text{\it SW}} \def\tbd{\text{\color{red} ?}} \def\bbs#1#2#3{{\href{http://drorbn.net/bbs/show?shot=#1-#2-#3.jpg}{BBS:\linebreak[0]#1-\linebreak[0]#2}}} \paperwidth 8in \paperheight 10.5in \textwidth 8in \textheight 10.5in \oddsidemargin -0.75in \evensidemargin \oddsidemargin \topmargin -0.75in \headheight 0in \headsep 0in \footskip 0in \parindent 0in \setlength{\topsep}{0pt} \def\cellscale{0.645} \pagestyle{empty} \dmyydate \newcounter{linecounter} \newcommand{\cheatline}{\vskip 1mm\noindent\refstepcounter{linecounter}\thelinecounter. } \begin{document} %\setlength{\jot}{0ex} \setlength{\abovedisplayskip}{0.5ex} \setlength{\belowdisplayskip}{0.5ex} \setlength{\abovedisplayshortskip}{0ex}%\setlength{\belowdisplayshortskip}{0ex} {\LARGE{\bf Cheat Sheet $sl_2$-Invariant}}\hfill(the $sl_2$ portfolio and invariant)\hfill \parbox[b]{2.5in}{\tiny \null\hfill\url{http://drorbn.net/AcademicPensieve/Projects/SL2Invariant/} \newline\null\hfill modified \today, \currenttime } \vskip -3mm \rule{\textwidth}{1pt} \vspace{-8mm} \begin{multicols}{2} \raggedcolumns {\red\bf Objects.} All are of the form $\bbe^{L+Q}P,ドル where \begin{itemize}[leftmargin=*,labelindent=0pt,itemsep=-2pt,topsep=0pt] \item $L$ is a quadratic of the form $\sum l_{z\zeta}z\zeta,ドル where $z$ runs over $\{t_i,\alpha_i\}_{i\in S}$ and $\zeta$ over $\{\tau_i,a_i\}_{i\in S},ドル with integer coefficients $l_{z\zeta}$. \item $Q$ is a quadratic of the form $\sum q_{z\zeta}z\zeta,ドル where $z$ runs over $\{x_i,\eta_i\}_{i\in S}$ and $\zeta$ over $\{\xi_i,y_i\}_{i\in S},ドル with coefficients $q_{z\zeta}$ in the ring $R_S$ of rational functions in $\{T_i,\calA_i\}_{i\in S}$. \item $P=\sum\epsilon^kP_k$ is docile ($\deg P_k\leq 4k$) in $\{y_i,a_i,x_i,\eta_i,\xi_i\}_{i\in S}$ with coefficients in $R_S,ドル and where $\deg(y_i,a_i,x_i,\eta_i,\xi_i)=(1,2,1,1,1)$. \end{itemize} {\red\bf In {\tt QuarksAndDegrees.m}:} Gradings to remove $\hbar$ and $\gamma$. {\red\bf Q.} What becomes of the classical-level automorphism $(y,b,\epsilon)\to(-y,-b,-\epsilon)$? \newcount\snip \snip=0\loop \advance \snip 1 \ifnum \snip=19\newline{ {\red\bf The Zipping Theorem.} If $P$ has a finite $\zeta$-degree, \begin{multline*} \left\langle P(z_i,\zeta^j)\bbe^{c+\eta^iz_i+y_j\zeta^j+q^i_jz_i\zeta^j}\right\rangle_{(\zeta^j)} \\ = |\tilde{q}|\bbe^{c+\eta^i\tilde{q}_i^ky_k} \left\langle P\left( \tilde{q}_i^k(z_k+y_k),\zeta^j+\eta^i\tilde{q}_i^j \right) \right\rangle_{(\zeta^j)}. \end{multline*} where $\tilq$ is the inverse matrix of 1ドル-q$: $(\delta^i_j-q^i_j)\tilq^j_k=\delta^i_k$. }\fi \ifnum \snip=38\newline\includegraphics[width=\linewidth]{RVKSummary.png} \fi \par\needspace{20mm}\includegraphics[scale=\cellscale]{Snips/Program-\the\snip.pdf} \ifnum \snip<40 \repeat \par\needspace{20mm}\includegraphics[scale=\cellscale]{Snips/Knot-1.pdf} \par\needspace{20mm}\includegraphics[scale=\cellscale]{Snips/Knot-2.pdf} \par\needspace{20mm}\includegraphics[scale=\cellscale]{Snips/Profile-1.pdf} \par\needspace{20mm}\includegraphics[height=\textheight]{Snips/Profile-2.pdf} \end{multicols} \end{document} \endinput

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