\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 $sl_2$-Portfolio}}\hfill(an implementation of the $sl_2$ portfolio)\hfill \parbox[b]{2.5in}{\tiny \null\hfill\url{http://drorbn.net/AcademicPensieve/Projects/SL2Portfolio/} \newline\null\hfill modified \today, \currenttime } \vskip -3mm \rule{\textwidth}{1pt} \vspace{-8mm} \begin{multicols}{2} \raggedcolumns %\entry{170625} {\bf $\calU_{\gamma\epsilon;\hbar}$ conventions.} \hfill \pinkt{``consolidate''} $\pinkm{q=\bbe^{\hbar\gamma\epsilon}}$, $H=\langle a,x\rangle/(\yellowm{[a,x]=\gamma x})$ with \[ \pinkm{A=\bbe^{-\hbar\epsilon a}}, \quad xA=qAx, \quad S_{\!H}(a,A,x)=(-a, A^{-1}, -A^{-1}x), \] \[ \Delta_H(a,A,x)=(a_1+a_2, A_1A_2, x_1+A_1x_2) \] and dual $H^\ast=\langle b, y\rangle/(\yellowm{[b,y]=-\epsilon y})$ with \[ \pinkm{B=\bbe^{-\hbar\gamma b}}, \quad By=qyB, \quad S_{\!H^\ast}(b,B,y)=(-b, B^{-1}, -yB^{-1}), \] \[ \Delta_{H^\ast}(b,B,y)=(b_1+b_2, B_1B_2, y_1B_2+y_2). \] Pairing by $(a,x)^\ast=\hbar(b,y)$ ($\Rightarrow\langle B,A\rangle=q$) making $\langle y^lb^i,a^jx^k\rangle = \delta_{ij}\delta_{kl}\hbar^{-(j+k)}j![k]_q!$ so $R=\sum\frac{\hbar^{j+k}y^kb^j\otimes a^jx^k}{j![k]_q!}$. Then $\calU=H^{\ast\text{\it cop}}\otimes H$ with $(\phi f)(\psi g) = \langle \psi_1S^{-1}f_3\rangle \langle \psi_3,f_1\rangle(\phi\psi_2)(f_2g)$ and \[ S(y,b,a,x) = (-B^{-1}y, -b, -a, -A^{-1}x),\] \[ \Delta(y,b,a,x) = (y_1+y_2B_1, b_1+b_2, a_1+a_2, x_1+A_1x_2).\] With the central $t\coloneqq\epsilon a-\gamma b$, $\pinkm{T\coloneqq \bbe^{\hbar t}}=A^{-1}B$ get \[ \yellowm{[a,y]=-\gamma y}, \quad \yellowm{[b,x]=\epsilon x},\quad \yellowm{xy-qyx=(1-TA^2)/\hbar}. \] Cartan: $\theta(y, b, a, x)=(-B^{-1}T^{1/2}x, -b, -a, -A^{-1}T^{-1/2}y)$. (Suggesting that it may be better to redefine $y\to y'=\theta x=A^{-1}T^{-1/2}y$.) At $\epsilon=0$, $\calU_{\hbar;\gamma 0}=\langle t,y,a,x\rangle/([t,\cdot]=0,\,[a,x]=\gamma x,\,[a,y]=-\gamma y,\,[x,y]=(1-T)/\hbar)$ with $\Delta(t,y,a,x)=(t_1+t_2,y_1+T_1y_2,a_1+a_2, x_1+x_2)$ and $\theta(y, b, a, x)=(-T^{-1/2}x, -b, -a, -T^{-1/2}y)$. {\bf Working Hypothesis.} $(\hbar,t,y,a,x)$ makes a PBW basis. {\bf Casimir.} $\omega=\gamma yx+\epsilon a^2-(t-\gamma\epsilon)a$, satisfies\ldots. Roland in \href{http://drorbn.net/AcademicPensieve/People/VanDerVeen/MixOrder@.pdf}{MixOrder.pdf}: Centrals are valuable; perhaps we should write everything in $CU/QU$ as $(x\vee y)\cdot$(functions of $a$)$\cdot$(centrals). {\bf Scaling} with $\deg\colon\{\gamma,\epsilon,a,b,x,y\}\to 1,\,\{\hbar\}\to -2,\,\{t\}\to 2,\,\{\omega\}\to 3$. \rule{\linewidth}{0.5pt} \vskip 2mm {\bf Verification} (as in \href{http://drorbn.net/AcademicPensieve/Projects/PPSA/nb/Verification.pdf}{Projects/PPSA/Verification.nb}). \vskip 1mm \par\includegraphics[scale=\cellscale]{Snips/DocileQ-1.pdf} \par\includegraphics[scale=\cellscale]{Snips/TD-1.pdf} \par\includegraphics[scale=\cellscale]{Snips/CF-1.pdf} \par\includegraphics[scale=\cellscale]{Snips/SeriesData-1.pdf} \par\includegraphics[scale=\cellscale]{Snips/SP-1.pdf} \par\includegraphics[scale=\cellscale]{Snips/CU-1.pdf} \par\includegraphics[scale=\cellscale]{Snips/QU-1.pdf} \par\includegraphics[scale=\cellscale]{Snips/theta-1.pdf} \par\includegraphics[scale=\cellscale]{Snips/ADeq-1.pdf} \par\includegraphics[scale=\cellscale]{Snips/ADeq-2.pdf} \par\includegraphics[scale=\cellscale]{Snips/ADeq-3.pdf} \newcount\snip \snip=1\loop \par\includegraphics[scale=\cellscale]{Snips/SDeq-\the\snip.pdf} \advance \snip 1 \ifnum \snip<5 \repeat \par\includegraphics[scale=\cellscale]{Snips/rho-1.pdf} {\bf Fear Not.} If $G=\bbe^{\xi x}y\bbe^{-\xi x}$ then $F = \bbe^{-\eta y}\bbe^{\xi x}\bbe^{\eta y}\bbe^{-\xi x} = \bbe^{-\eta y}\bbe^{\eta G}$ satisfies $\partial_\eta F=-yF+FG$ and $F_{\eta=0}=1$: \par\includegraphics[scale=\cellscale]{Snips/tSW-1.pdf} \par\includegraphics[scale=\cellscale]{Snips/Faddeev-1.pdf} \par\includegraphics[scale=\cellscale]{Snips/R-1.pdf} \par\includegraphics[scale=\cellscale]{Snips/Exp-1.pdf} \par\includegraphics[scale=\cellscale]{Snips/Exp-2.pdf} \rule{\linewidth}{0.5pt} \vskip 1mm {\bf To do.} $\bullet$~Consider renormalizing $x$ and $y$. $\bullet$~Can everything be done at $\hbar=1$ defining a filtration by other means? That ought to be possible as the end results depend on $t/T$ and not on $\hbar$. $\bullet$ Bound the degrees of the logoi! $\bullet$ $r=\theta r$? $\bullet$ $\theta$ is a global symmetry. Can it be ``gauged''? $\bullet$ Global $\eta\to\psi$? \rule{\linewidth}{0.5pt} {\bf Alternative Algorithms.} \vskip 1mm \par\includegraphics[scale=\cellscale]{Snips/AltLogos-1.pdf} \vskip 1mm{\bf Asides.} \quad \includegraphics[scale=\cellscale]{Snips/Aside-1.pdf} \vskip 2mm\par\includegraphics[scale=\cellscale]{Snips/Aside-2.pdf} \rule{\linewidth}{0.5pt} \vskip 1mm {\bf $\calG\calD\calO$-Categories.} Given $\frakg$ with basis $B=\{x,y,\ldots\}$, consider the following diagram: \[ \xymatrix@C=11.5mm{ \bbQ=\hat\calU_{(q)}\left(\bigoplus_0\frakg\right) \ar[r]^<>(0.5)Z & \hat\calU_{(q)}\left(\frakg\right) \ar@/^/[r]^<>(0.5)\Delta & \hat\calU_{(q)}\left(\bigoplus_2\frakg\right) \ar@/^/[l]^<>(0.5)m \\ \hat\calS\left(\emptyset\right) \ar[r]^<>(0.5)Z \ar[u] & \hat\calS\left(B\right) \ar@/^/[r]^<>(0.5)\Delta \ar@/^/[u]^{\bbO(xy\ldots\colon\cdot)} \ar@/_/[u]_{\bbO(yx\ldots\colon\cdot)} \ar@`{p+(-16,-16),p+(+16,-16)}^{\SW_{xy}} & \hat\calS\left(B_1,B_2\right) \ar@/^/[l]^<>(0.5)m \ar[u]_{\bbO(y_1x_1\ldots\otimes y_2x_2\ldots\colon\cdot)} } \] Hence $Z$, $\SW_{xy}$, $m$, $\Delta$, (and likewise $S$ and $\theta$) are morphisms in the {\em completion} of the monoidal category $\calF$ whose objects are finite sets $B$ and whose morphism are $\mor_\calF(B,B') \coloneqq \Hom_\bbQ\left(\calS(B)\to\calS(B')\right) = \calS\left(B^\ast,B'\right)$ (by convention, $x^\ast=\xi$, $y^\ast=\eta$, etc.). Ergo we need to {\em consolidate} (at least parts of) said completion. {\bf Aside.} ``Consolidate'' means ``give a finite name to an infinite object, and figure out how to sufficiently manipulate such finite names''. E.g., solving $f''=-f$ we encounter and set $\sum\frac{(-1)^kx^{2k}}{(2k)!}\leadsto\cos x$, $\sum\frac{(-1)^kx^{2k+1}}{(2k+1)!}\leadsto\sin x$, and then $\cos^2x+\sin^2x=1$ and $\sin(x+y)=\sin x\cos y+\cos x\sin y$. {\bf Example.} {\bf Example.} In $QU/(\epsilon^2=0)$ using the $yax$ order over $\bbQ\llbracket\hbar\rrbracket$, with $T=\bbe^{\hbar t}$, $\barT=T^{-1}$, $\calA=\bbe^{\gamma\alpha}$, and $\bcA=\calA^{-1}$, \[ R_{ij} = \bbe^{\hbar(y_ix_j-t_ia_j/\gamma)} \left(1+\epsilon\hbar\left(a_ia_j/\gamma-\gamma\hbar^2y_i^2x_j^2/4\right)\right) \in \calS(B_i,B_j), \] \begin{multline*} m = \bbe^{(\alpha _1+\alpha _2)a+\eta _2 \xi _1(1-T)/\hbar + (\xi_1\bcA_2 + \xi _2)x + (\eta_1+\eta_2\bcA_1) y} \left(1+\epsilon\lambda_m\right) \\ \in \calS(B^\ast_1,B^\ast_2,B), \end{multline*} with {\footnotesize$\lambda_m = 2a\eta_2\xi_1T + \frac{1}{4}\gamma\eta_2^2\xi_1^2\left(3T^2-4T+1\right)/\hbar - \frac{1}{2}\gamma\eta_2\xi_1^2(3T-1)x\bcA_2 - \frac{1}{2}\gamma\eta_2^2\xi_1(3 T-1)y\bcA_1 + \gamma\eta_2\xi_1xy\hbar\bcA_1\bcA_2$}, \begin{multline*} \Delta = \bbe^{\tau(t_1+t_1)+\eta(y_1+T_1y_2)+\alpha(a_1+a_2)+\xi(x_1+x_2)}\left(1 + \epsilon\lambda_\Delta\right) \\ \in \calS(B^\ast,B_1,B_2), \end{multline*} with {\scriptsize $\lambda_\Delta = -a_1\eta T_1y_2\hbar - a_1\xi x_2\hbar + \frac{1}{2}\gamma\eta^2T_1y_1y_2\hbar + \frac{1}{2}\gamma\xi^2 x_1 x_2 \hbar$}, and \[ S = \bbe^{-\tau t -\alpha a -\eta \xi (1-\barT)\calA/\hbar - \barT\eta y \calA -\xi x \calA}\left(1+\epsilon \lambda_S\right) \in\calS(B^\ast,B), \] with {\footnotesize$\lambda_S = 2\barT\calA a \eta \xi - \barT\calA a \eta y \hbar - a \xi x \hbar \calA - \frac14 \gamma \eta ^2 \xi ^2 \left(1-4\barT+3\barT^2\right)\calA^2/\hbar - \frac12\gamma \eta ^2 y^2 \hbar\barT^2\calA^2 - \frac12\gamma\eta^2\xi\barT(1-3\barT) y \calA^2 + \gamma\eta \xi(1-\barT)\calA - \frac12\gamma\eta\xi^2(1-3\barT) x \calA^2 - \gamma \eta \xi x y \hbar \barT\calA^2 + \gamma \eta y \hbar\barT\calA - \frac{1}{2} \gamma \xi x^2 \hbar \calA^2 $}. {\bf Problem.} Compute the likes of $m\act\Delta = \left(\left.m\right|_{b\to\partial_\beta}\Delta\right)_{\beta=0}$ and $(R_{12}R_{34})\act m^{13}_2 = \left(\left.(R_{12}R_{34})\right|_{b\to\partial_\beta}m^{13}_2\right)_{\beta=0}$. A generic morphism: \[ \input{GDOMorphism.pdf_t} \] {\bf The Zipping Issue.}\hfill\includegraphics[width=0.8in]{zipper.png}\hfill\null {\bf The Contraction Theorem.} If $P$ has a finite $\zeta$-degree and the $y$'s and the $q$'s are ``small'', \[ \left\langle P(z_i,\zeta^j)\right\rangle_{(\zeta_i)} = \left.P\left(z_i,\overset{\leftrightarrow}{\partial}_{\!z_j}\right)\right|_{z_i=0}, \] \[ \left\langle P(z_i,\zeta^j)\bbe^{\eta^iz_i+y_j\zeta^j}\right\rangle_{(\zeta_i)} = \left\langle P(z_i+y_i,\zeta^j)\bbe^{\eta^i(z_i+y_i)}\right\rangle_{(\zeta_i)}, \] (proof: replace $y_j\to\hbar y_j$ and test at $\hbar=0$ and at $\partial_\hbar$), and \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_i)} \\ = \det(\tilq)\left\langle P(\tilq_i^k(z_k+y_k),\zeta^j)\bbe^{c+\eta^i\tilq^k_i(z_k+y_k)} \right\rangle_{(\zeta_i)} \end{multline*} where $\tilq$ is the inverse matrix of $1-q$: $(\delta^i_j-q^i_j)\tilq^j_k=\delta^i_k$ (proof: replace $q^i_j\to\hbar q^i_j$ and test at $\hbar=0$ and at $\partial_\hbar$). \par\includegraphics[scale=\cellscale]{Snips/E-1.pdf} \par\includegraphics[scale=\cellscale]{Snips/Zip-1.pdf} \par\includegraphics[scale=\cellscale]{Snips/Zip-2.pdf} \par\includegraphics[scale=\cellscale]{Snips/Zip-3.pdf} \par\includegraphics[scale=\cellscale]{Snips/Zip-4.pdf} \par\includegraphics[scale=\cellscale]{Snips/Bind-1.pdf} \par\includegraphics[scale=\cellscale]{Snips/t1-1.pdf} \par\includegraphics[scale=\cellscale]{Snips/tm-1.pdf} \par\includegraphics[scale=\cellscale]{Snips/tS-1.pdf} \par\includegraphics[scale=\cellscale]{Snips/tDelta-1.pdf} \par\includegraphics[scale=\cellscale]{Snips/tR-1.pdf} \par\includegraphics[scale=\cellscale]{Snips/tC-1.pdf} \par\includegraphics[scale=\cellscale]{Snips/tKink-1.pdf} %\end{multicols} %\newpage %\begin{multicols}{2} \rule{\linewidth}{0.5pt} \vskip 1mm {\bf Program} (as in \href{http://drorbn.net/AcademicPensieve/Projects/PPSA/nb/Verification.pdf}{Projects/PPSA/Verification.nb}). \vskip 1mm %\newcount\snip %\snip=1\loop % \par\includegraphics[scale=\cellscale]{Snips/QLImplementation-\the\snip.pdf} % \advance \snip 1 %\ifnum \snip<5 \repeat \par\includegraphics[scale=\cellscale]{Snips/QLImplementation-1.pdf} \par\includegraphics[scale=\cellscale]{Snips/QLImplementation-3.pdf} \par\includegraphics[scale=\cellscale]{Snips/QLImplementation-4.pdf} \par\includegraphics[scale=\cellscale]{Snips/QLImplementation-2.pdf} %\vfill\par\includegraphics[width=\columnwidth]{../../People/ThurstonD/Agenda.png} \end{multicols} \end{document} \endinput