\documentclass[11pt,notitlepage]{article}
\usepackage{amsmath, graphicx, amssymb, stmaryrd, datetime, txfonts, multicol, calc, import, amscd, picins, enumitem, wasysym, needspace, mathbbol, import}
\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{\mor}{\operatorname{mor}}
\def\sder{\operatorname{\mathfrak{sder}}}

\def\act{{\hspace{-1pt}\sslash\hspace{-0.75pt}}}
\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\calA{{\mathcal A}}
\def\calD{{\mathcal D}}
\def\calL{{\mathcal L}}
\def\calP{{\mathcal P}}
\def\calS{{\mathcal S}}
\def\calU{{\mathcal U}}
\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\frakb{{\mathfrak b}}
\def\frakg{{\mathfrak g}}
\def\frakt{{\mathfrak t}}
\def\Loneco{{\calL^{\text{1co}}}}
\def\PaT{\text{{\bf PaT}}}
\def\PvT{{\mathit P\!v\!T}}
\def\remove{\!\setminus\!}
\def\tbd{\text{\color{red} ?}}
\def\tder{\operatorname{\mathfrak{tder}}}
\def\TAut{\operatorname{TAut}}

\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{0.662}
\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 PPSA}}\hfill(formulas for the PPSA paper)\hfill
\parbox[b]{2.5in}{\tiny
  \null\hfill\url{http://drorbn.net/AcademicPensieve/Projects/PPSA/}
  \newline\null\hfill
   modified \today.
}

\vskip -3mm
\rule{\textwidth}{1pt}
\vspace{-8mm}

\begin{multicols}{2}

%\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 $t\coloneqq\epsilon a-\gamma b$, $\pinkm{T\coloneqq e^{\hbar t/2}}=A^{-1/2}B^{1/2}$ get
\[ \yellowm{[a,y]=-\gamma y}, \quad \yellowm{[b,x]=\epsilon x},\quad \yellowm{xy-qyx=(1-T^2A^2)/\hbar}. \]
Cartan: $\theta(y, b, a, x)=(-B^{-1}Tx, -b, -a, -A^{-1}T^{-1}y)$. (Suggesting that it may be better to redefine $y\to y'=\theta x=A^{-1}T^{-1}y$.)

At $\epsilon=0$, $\calU_{\hbar;\gamma 0}=\langle b,y,a,x\rangle/([b,\cdot]=0,\,[a,x]=\gamma x,\,[a,y]=-\gamma y,\,[x,y]=(1-\bbe^{-\hbar\gamma b})/\hbar)$ with $\Delta(b,y,a,x)=(b_1+b_2,y_1+\bbe^{-\hbar\gamma b_1}y_2,a_1+a_2, x_1+x_2)$ and $\theta(y, b, a, x)=(-\bbe^{\hbar\gamma b/2}x, -b, -a, -\bbe^{\hbar\gamma b/2}y)$.

%Benkart-Witherspoon, \href{http://drorbn.net/AcademicPensieve/2017-06/nb/BW.pdf}{2017-06/BW.nb}: At $\gamma\epsilon\hbar=\sigma-\rho$, represented by
%$y\to\begin{pmatrix}0&0\\-\bbe^\rho&0\end{pmatrix}$,
%$a\to\frac{\gamma}{\rho-\sigma}\begin{pmatrix}\rho&0\\0&\sigma\end{pmatrix}$,
%$A\to\begin{pmatrix}\bbe^\rho&0\\0&\bbe^\sigma\end{pmatrix}$,
%$x\to\frac{\bbe^\rho-\bbe^\sigma}{\hbar \bbe^{\rho+\sigma}}\begin{pmatrix}0&1\\0&0\end{pmatrix}$,
%$t\to\frac{\rho+\sigma}{\hbar}\begin{pmatrix}1&0\\0&1\end{pmatrix}$,
%$T\to\bbe^{-(\rho+\sigma)/2}\begin{pmatrix}1&0\\0&1\end{pmatrix}$,
%$b\to\frac{\epsilon}{\sigma-\rho}\begin{pmatrix}\sigma&0\\0&\rho\end{pmatrix}$,
%$B\to\begin{pmatrix}\bbe^{-\sigma}&0\\0&\bbe^{-\rho}\end{pmatrix}$.

{\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

{\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]{VSnips/TD-1.pdf}

\par\includegraphics[scale=\cellscale]{VSnips/Utils-1.pdf}

\par\includegraphics[scale=\cellscale]{VSnips/CU-1.pdf}

\par\includegraphics[scale=\cellscale]{VSnips/QU-1.pdf}

\par\includegraphics[scale=\cellscale]{VSnips/theta-1.pdf}

Can the $A\bbD$ and $S\bbD$ formulas be written so as to manifestly see their lowest term in $\epsilon$? This may allow more flexibility with {\tt \$T$\epsilon$D}. Or perhaps better, these should be written in implicit form and solved by power series.

\par\includegraphics[scale=\cellscale]{VSnips/ADeq-1.pdf}
\par\includegraphics[scale=\cellscale]{VSnips/ADeq-2.pdf}
\par\includegraphics[scale=\cellscale]{VSnips/ADeq-3.pdf}

\newcount\snip
\snip=1\loop
  \par\includegraphics[scale=\cellscale]{VSnips/SDeq-\the\snip.pdf}
  \advance \snip 1
\ifnum \snip<5 \repeat

\par\includegraphics[scale=\cellscale]{VSnips/Quesne-1.pdf}

\par\includegraphics[scale=\cellscale]{VSnips/R-1.pdf}

\par\includegraphics[scale=\cellscale]{VSnips/CdsO-1.pdf}

\par\includegraphics[scale=\cellscale]{VSnips/rho-1.pdf}

\par\includegraphics[scale=\cellscale]{VSnips/Logos-1.pdf}
\par\includegraphics[scale=\cellscale]{VSnips/Logos-2.pdf}

\newcount\snip
\snip=1\loop
  \par\includegraphics[scale=\cellscale]{VSnips/SW-\the\snip.pdf}
  \advance \snip 1
\ifnum \snip<3 \repeat

\rule{\linewidth}{0.5pt}

\vskip 2mm
{\bf To do.}
$\bullet$~Consider renormalizing $x$ and $y$.
$\bullet$~Implement variable swaps.
$\bullet$~Implement $m_{ij\to k}$.
$\bullet$~Implement $\bbE$, $R\bbE$, and the casts {\tt CU} and $\tt QU$.
$\bullet$~Reconsider the expansion of $T$ and $q$ in the hope of improving speed.

%$\bullet$~I should re-implement the $\hbar^{>d}$-bound within {\tt Simp/CE}, so as to eliminate the need for {\tt SS} and for the expansion of $T$.

\end{multicols}
\newpage
\begin{multicols}{2}

{\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]{VSnips/QLImplementation-\the\snip.pdf}
  \advance \snip 1
\ifnum \snip<5 \repeat

\rule{\linewidth}{0.5pt}

\vfill

\par\includegraphics[width=\columnwidth]{../../People/ThurstonD/Agenda.png}

\end{multicols}

\end{document}

\endinput